Embedded newform 43.3.h.a.12.5

• Label: 43.3.h.a.12.5
• Level: $43$
• Weight: $3$
• Character: 43.12
• 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			12.5
Character	\(\chi\)	\(=\)	43.12
Dual form			43.3.h.a.18.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.704959 + 1.46386i) q^{2} +(0.920816 + 1.35059i) q^{3} +(0.848033 - 1.06340i) q^{4} +(0.650793 - 0.701388i) q^{5} +(-1.32794 + 2.30006i) q^{6} +(-11.1429 + 6.43335i) q^{7} +(8.49061 + 1.93793i) q^{8} +(2.31188 - 5.89057i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 14 q^{2} - 14 q^{3} + 12 q^{4} - 11 q^{5} + 2 q^{6} - 30 q^{7} - 42 q^{8} + 54 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{13}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.704959	+	1.46386i	0.352480	+	0.731931i	0.999534	−	0.0305121i	\(-0.00971382\pi\)
							−0.647055	+	0.762443i	\(0.724000\pi\)
\(3\)	0.920816	+	1.35059i	0.306939	+	0.450196i	0.948454	−	0.316913i	\(-0.102646\pi\)
							−0.641516	+	0.767110i	\(0.721694\pi\)
\(5\)	0.650793	−	0.701388i	0.130159	−	0.140278i	−0.664609	−	0.747191i	\(-0.731402\pi\)
							0.794768	+	0.606913i	\(0.207593\pi\)
\(7\)	−11.1429	+	6.43335i	−1.59184	+	0.919049i	−0.598849	+	0.800862i	\(0.704375\pi\)
							−0.992991	+	0.118188i	\(0.962292\pi\)
\(11\)	−10.6526	−	13.3579i	−0.968417	−	1.21436i	−0.976748	−	0.214392i	\(-0.931223\pi\)
							0.00833025	−	0.999965i	\(-0.497348\pi\)
\(13\)	14.6056	−	4.50523i	1.12351	−	0.346556i	0.323310	−	0.946293i	\(-0.395204\pi\)
							0.800197	+	0.599737i	\(0.204728\pi\)
\(17\)	−13.4977	+	12.5241i	−0.793985	+	0.736710i	−0.968787	−	0.247893i	\(-0.920262\pi\)
							0.174802	+	0.984604i	\(0.444071\pi\)
\(19\)	9.57253	−	3.75694i	0.503817	−	0.197734i	−0.0998006	−	0.995007i	\(-0.531820\pi\)
							0.603618	+	0.797274i	\(0.293725\pi\)
\(23\)	−24.6616	+	3.71715i	−1.07225	+	0.161615i	−0.661358	−	0.750071i	\(-0.730019\pi\)
							−0.410888	+	0.911686i	\(0.634781\pi\)
\(29\)	−6.56504	+	9.62915i	−0.226381	+	0.332040i	−0.922578	−	0.385811i	\(-0.873922\pi\)
							0.696197	+	0.717851i	\(0.254874\pi\)
\(31\)	−1.44091	+	19.2277i	−0.0464811	+	0.620247i	0.924514	+	0.381148i	\(0.124471\pi\)
							−0.970995	+	0.239099i	\(0.923148\pi\)
\(37\)	13.2991	+	7.67825i	0.359436	+	0.207520i	0.668833	−	0.743412i	\(-0.266794\pi\)
							−0.309398	+	0.950933i	\(0.600127\pi\)
\(41\)	13.5960	−	6.54750i	0.331610	−	0.159695i	−0.260668	−	0.965428i	\(-0.583943\pi\)
							0.592278	+	0.805733i	\(0.298229\pi\)
\(43\)	−19.6767	+	38.2338i	−0.457598	+	0.889159i
							\(47\)	1.51781	−	1.90327i	0.0322938	−	0.0404952i	−0.765422	−	0.643528i	\(-0.777470\pi\)
0.797716	+	0.603033i	\(0.206041\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.h.a.12.5	✓	72
3.2	odd	2		387.3.bn.b.55.2		72
43.18	odd	42	inner	43.3.h.a.18.5	yes	72
129.104	even	42		387.3.bn.b.190.2		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.h.a.12.5	✓	72	1.1	even	1	trivial
43.3.h.a.18.5	yes	72	43.18	odd	42	inner
387.3.bn.b.55.2		72	3.2	odd	2
387.3.bn.b.190.2		72	129.104	even	42