Embedded newform 43.3.h.a.5.6

• Label: 43.3.h.a.5.6
• Level: $43$
• Weight: $3$
• Character: 43.5
• Analytic conductor: $1.172$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.17166513675\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(6\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			5.6
Character	\(\chi\)	\(=\)	43.5
Dual form			43.3.h.a.26.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.50298 + 0.571289i) q^{2} +(0.445320 - 1.44369i) q^{3} +(2.33467 + 1.12432i) q^{4} +(-3.49724 + 1.37257i) q^{5} +(1.93939 - 3.35913i) q^{6} +(-3.35697 + 1.93815i) q^{7} +(-2.82762 - 2.25495i) q^{8} +(5.55021 + 3.78407i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 14 q^{2} - 14 q^{3} + 12 q^{4} - 11 q^{5} + 2 q^{6} - 30 q^{7} - 42 q^{8} + 54 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{25}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.50298	+	0.571289i	1.25149	+	0.285645i	0.796383	−	0.604792i	\(-0.206744\pi\)
							0.455107	+	0.890437i	\(0.349601\pi\)
\(3\)	0.445320	−	1.44369i	0.148440	−	0.481231i	−0.850563	−	0.525873i	\(-0.823739\pi\)
							0.999003	+	0.0446425i	\(0.0142149\pi\)
\(5\)	−3.49724	+	1.37257i	−0.699449	+	0.274513i	−0.688302	−	0.725425i	\(-0.741643\pi\)
							−0.0111469	+	0.999938i	\(0.503548\pi\)
\(7\)	−3.35697	+	1.93815i	−0.479568	+	0.276879i	−0.720236	−	0.693729i	\(-0.755967\pi\)
							0.240669	+	0.970607i	\(0.422633\pi\)
\(11\)	−2.36869	+	1.14070i	−0.215335	+	0.103700i	−0.538442	−	0.842662i	\(-0.680987\pi\)
							0.323107	+	0.946362i	\(0.395273\pi\)
\(13\)	4.94550	+	0.745414i	0.380423	+	0.0573396i	0.336470	−	0.941694i	\(-0.390767\pi\)
							0.0439528	+	0.999034i	\(0.486005\pi\)
\(17\)	8.80586	−	22.4370i	0.517992	−	1.31982i	−0.397543	−	0.917583i	\(-0.630137\pi\)
							0.915535	−	0.402238i	\(-0.131768\pi\)
\(19\)	3.29776	+	4.83693i	0.173566	+	0.254575i	0.903158	−	0.429308i	\(-0.141242\pi\)
							−0.729592	+	0.683883i	\(0.760290\pi\)
\(23\)	−0.176271	+	2.35218i	−0.00766397	+	0.102269i	−0.999724	−	0.0234975i	\(-0.992520\pi\)
							0.992060	+	0.125766i	\(0.0401389\pi\)
\(29\)	9.16398	+	29.7089i	0.315999	+	1.02444i	0.964689	+	0.263392i	\(0.0848412\pi\)
							−0.648690	+	0.761053i	\(0.724683\pi\)
\(31\)	−4.20063	−	3.89761i	−0.135504	−	0.125729i	0.609509	−	0.792779i	\(-0.291366\pi\)
							−0.745013	+	0.667049i	\(0.767557\pi\)
\(37\)	59.7484	+	34.4957i	1.61482	+	0.932317i	0.988231	+	0.152968i	\(0.0488831\pi\)
							0.626590	+	0.779349i	\(0.284450\pi\)
\(41\)	7.95064	−	34.8340i	0.193918	−	0.849611i	−0.780552	−	0.625091i	\(-0.785062\pi\)
							0.974470	−	0.224519i	\(-0.0720812\pi\)
\(43\)	−41.1003	+	12.6399i	−0.955820	+	0.293951i
							\(47\)	−68.9878	−	33.2228i	−1.46783	−	0.706868i	−0.482240	−	0.876039i	\(-0.660177\pi\)
−0.985587	+	0.169171i	\(0.945891\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.h.a.5.6	✓	72
3.2	odd	2		387.3.bn.b.91.1		72
43.26	odd	42	inner	43.3.h.a.26.6	yes	72
129.26	even	42		387.3.bn.b.370.1		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.h.a.5.6	✓	72	1.1	even	1	trivial
43.3.h.a.26.6	yes	72	43.26	odd	42	inner
387.3.bn.b.91.1		72	3.2	odd	2
387.3.bn.b.370.1		72	129.26	even	42