Embedded newform 43.3.h.a.18.2

• Label: 43.3.h.a.18.2
• Level: $43$
• Weight: $3$
• Character: 43.18
• 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			18.2
Character	\(\chi\)	\(=\)	43.18
Dual form			43.3.h.a.12.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.963431 + 2.00058i) q^{2} +(1.57068 - 2.30376i) q^{3} +(-0.580179 - 0.727522i) q^{4} +(4.59096 + 4.94788i) q^{5} +(3.09563 + 5.36179i) q^{6} +(-4.41636 - 2.54979i) q^{7} +(-6.64480 + 1.51663i) q^{8} +(0.447774 + 1.14091i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 14 q^{2} - 14 q^{3} + 12 q^{4} - 11 q^{5} + 2 q^{6} - 30 q^{7} - 42 q^{8} + 54 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{29}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.963431	+	2.00058i	−0.481715	+	1.00029i	0.508541	+	0.861038i	\(0.330185\pi\)
							−0.990257	+	0.139254i	\(0.955529\pi\)
\(3\)	1.57068	−	2.30376i	0.523560	−	0.767921i	−0.469824	−	0.882760i	\(-0.655683\pi\)
							0.993384	+	0.114839i	\(0.0366351\pi\)
\(5\)	4.59096	+	4.94788i	0.918192	+	0.989576i	0.999975	−	0.00707465i	\(-0.00225195\pi\)
							−0.0817831	+	0.996650i	\(0.526061\pi\)
\(7\)	−4.41636	−	2.54979i	−0.630909	−	0.364255i	0.150195	−	0.988656i	\(-0.452010\pi\)
							−0.781104	+	0.624401i	\(0.785343\pi\)
\(11\)	12.0209	−	15.0738i	1.09281	−	1.37034i	0.169843	−	0.985471i	\(-0.445674\pi\)
							0.922970	−	0.384873i	\(-0.125755\pi\)
\(13\)	−8.17956	−	2.52306i	−0.629197	−	0.194082i	−0.0362754	−	0.999342i	\(-0.511549\pi\)
							−0.592921	+	0.805260i	\(0.702026\pi\)
\(17\)	−3.38141	−	3.13749i	−0.198907	−	0.184558i	0.574431	−	0.818553i	\(-0.305224\pi\)
							−0.773337	+	0.633995i	\(0.781414\pi\)
\(19\)	−25.7368	−	10.1009i	−1.35457	−	0.531629i	−0.426651	−	0.904416i	\(-0.640307\pi\)
							−0.927917	+	0.372787i	\(0.878402\pi\)
\(23\)	−10.0951	−	1.52159i	−0.438916	−	0.0661559i	−0.0741326	−	0.997248i	\(-0.523619\pi\)
							−0.364783	+	0.931092i	\(0.618857\pi\)
\(29\)	3.91843	+	5.74728i	0.135118	+	0.198182i	0.887834	−	0.460163i	\(-0.152209\pi\)
							−0.752716	+	0.658345i	\(0.771257\pi\)
\(31\)	1.86051	+	24.8268i	0.0600166	+	0.800865i	0.943282	+	0.331992i	\(0.107721\pi\)
							−0.883266	+	0.468873i	\(0.844660\pi\)
\(37\)	7.06883	−	4.08119i	0.191049	−	0.110302i	−0.401424	−	0.915892i	\(-0.631485\pi\)
							0.592474	+	0.805590i	\(0.298151\pi\)
\(41\)	67.5492	+	32.5300i	1.64754	+	0.793414i	0.999494	+	0.0318139i	\(0.0101284\pi\)
							0.648047	+	0.761600i	\(0.275586\pi\)
\(43\)	−27.3837	+	33.1531i	−0.636831	+	0.771003i
							\(47\)	21.2788	+	26.6828i	0.452740	+	0.567718i	0.954851	−	0.297085i	\(-0.0960145\pi\)
−0.502111	+	0.864803i	\(0.667443\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.h.a.18.2	yes	72
3.2	odd	2		387.3.bn.b.190.5		72
43.12	odd	42	inner	43.3.h.a.12.2	✓	72
129.98	even	42		387.3.bn.b.55.5		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.h.a.12.2	✓	72	43.12	odd	42	inner
43.3.h.a.18.2	yes	72	1.1	even	1	trivial
387.3.bn.b.55.5		72	129.98	even	42
387.3.bn.b.190.5		72	3.2	odd	2