Embedded newform 43.5.h.a.29.8

• Label: 43.5.h.a.29.8
• Level: $43$
• Weight: $5$
• Character: 43.29
• 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			29.8
Character	\(\chi\)	\(=\)	43.29
Dual form			43.5.h.a.3.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.317475 - 0.659244i) q^{2} +(14.0677 - 1.05423i) q^{3} +(9.64202 - 12.0907i) q^{4} +(13.2386 + 42.9184i) q^{5} +(-5.16114 - 8.93936i) q^{6} +(-57.8093 - 33.3762i) q^{7} +(-22.4456 - 5.12306i) q^{8} +(116.694 - 17.5888i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q - 14 q^{2} - 20 q^{3} + 220 q^{4} - 11 q^{5} - 22 q^{6} - 150 q^{7} + 322 q^{8} - 660 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{41}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.317475	−	0.659244i	−0.0793688	−	0.164811i	0.857508	−	0.514471i	\(-0.172012\pi\)
							−0.936877	+	0.349660i	\(0.886297\pi\)
\(3\)	14.0677	−	1.05423i	1.56308	−	0.117137i	0.735125	−	0.677932i	\(-0.237124\pi\)
							0.827954	+	0.560795i	\(0.189505\pi\)
\(5\)	13.2386	+	42.9184i	0.529543	+	1.71674i	0.684680	+	0.728844i	\(0.259942\pi\)
							−0.155137	+	0.987893i	\(0.549582\pi\)
\(7\)	−57.8093	−	33.3762i	−1.17978	−	0.681147i	−0.223817	−	0.974631i	\(-0.571852\pi\)
							−0.955964	+	0.293484i	\(0.905185\pi\)
\(11\)	37.7924	+	47.3902i	0.312334	+	0.391654i	0.913077	−	0.407788i	\(-0.133700\pi\)
							−0.600743	+	0.799442i	\(0.705128\pi\)
\(13\)	14.0167	+	13.0056i	0.0829389	+	0.0769561i	0.720553	−	0.693400i	\(-0.243888\pi\)
							−0.637614	+	0.770356i	\(0.720079\pi\)
\(17\)	−299.351	−	92.3374i	−1.03582	−	0.319507i	−0.270199	−	0.962804i	\(-0.587090\pi\)
							−0.765616	+	0.643298i	\(0.777566\pi\)
\(19\)	88.7442	−	588.780i	0.245829	−	1.63097i	−0.436346	−	0.899779i	\(-0.643728\pi\)
							0.682175	−	0.731189i	\(-0.261034\pi\)
\(23\)	−258.307	+	658.155i	−0.488293	+	1.24415i	0.448348	+	0.893859i	\(0.352013\pi\)
							−0.936641	+	0.350291i	\(0.886083\pi\)
\(29\)	177.562	+	13.3065i	0.211132	+	0.0158222i	0.179876	−	0.983689i	\(-0.442430\pi\)
							0.0312560	+	0.999511i	\(0.490049\pi\)
\(31\)	617.761	+	421.182i	0.642831	+	0.438275i	0.840365	−	0.542021i	\(-0.182341\pi\)
							−0.197534	+	0.980296i	\(0.563293\pi\)
\(37\)	−149.252	+	86.1707i	−0.109023	+	0.0629442i	−0.553520	−	0.832836i	\(-0.686716\pi\)
							0.444497	+	0.895780i	\(0.353382\pi\)
\(41\)	−583.852	+	281.168i	−0.347324	+	0.167262i	−0.599410	−	0.800442i	\(-0.704598\pi\)
							0.252085	+	0.967705i	\(0.418884\pi\)
\(43\)	−832.112	+	1651.18i	−0.450033	+	0.893012i
							\(47\)	1919.00	−	2406.35i	0.868720	−	1.08934i	−0.126528	−	0.991963i	\(-0.540383\pi\)
0.995248	−	0.0973770i	\(-0.0310453\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.5.h.a.29.8	yes	168
43.3	odd	42	inner	43.5.h.a.3.8	✓	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.h.a.3.8	✓	168	43.3	odd	42	inner
43.5.h.a.29.8	yes	168	1.1	even	1	trivial