Embedded newform 43.3.h.a.3.6

• Label: 43.3.h.a.3.6
• Level: $43$
• Weight: $3$
• Character: 43.3
• 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			3.6
Character	\(\chi\)	\(=\)	43.3
Dual form			43.3.h.a.29.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.31976 - 2.74051i) q^{2} +(1.72784 + 0.129483i) q^{3} +(-3.27466 - 4.10629i) q^{4} +(-1.51671 + 4.91705i) q^{5} +(2.63518 - 4.56426i) q^{6} +(-4.74658 + 2.74044i) q^{7} +(-3.71320 + 0.847514i) q^{8} +(-5.93082 - 0.893928i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 14 q^{2} - 14 q^{3} + 12 q^{4} - 11 q^{5} + 2 q^{6} - 30 q^{7} - 42 q^{8} + 54 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.31976	−	2.74051i	0.659879	−	1.37025i	−0.255164	−	0.966898i	\(-0.582129\pi\)
							0.915043	−	0.403356i	\(-0.132156\pi\)
\(3\)	1.72784	+	0.129483i	0.575946	+	0.0431612i	0.359519	−	0.933138i	\(-0.382941\pi\)
							0.216427	+	0.976299i	\(0.430560\pi\)
\(5\)	−1.51671	+	4.91705i	−0.303342	+	0.983409i	0.667686	+	0.744443i	\(0.267285\pi\)
							−0.971028	+	0.238966i	\(0.923191\pi\)
\(7\)	−4.74658	+	2.74044i	−0.678083	+	0.391492i	−0.799133	−	0.601155i	\(-0.794707\pi\)
							0.121049	+	0.992647i	\(0.461374\pi\)
\(11\)	0.359494	−	0.450791i	0.0326813	−	0.0409810i	−0.765222	−	0.643766i	\(-0.777371\pi\)
							0.797904	+	0.602785i	\(0.205942\pi\)
\(13\)	10.4816	−	9.72551i	0.806277	−	0.748116i	−0.164942	−	0.986303i	\(-0.552744\pi\)
							0.971219	+	0.238187i	\(0.0765532\pi\)
\(17\)	1.14154	−	0.352118i	0.0671493	−	0.0207128i	−0.260999	−	0.965339i	\(-0.584052\pi\)
							0.328148	+	0.944626i	\(0.393576\pi\)
\(19\)	4.69930	+	31.1778i	0.247332	+	1.64094i	0.675586	+	0.737281i	\(0.263891\pi\)
							−0.428254	+	0.903658i	\(0.640871\pi\)
\(23\)	−15.3934	−	39.2217i	−0.669278	−	1.70529i	−0.708163	−	0.706049i	\(-0.750476\pi\)
							0.0388844	−	0.999244i	\(-0.487620\pi\)
\(29\)	−21.3574	+	1.60051i	−0.736461	+	0.0551901i	−0.437679	−	0.899131i	\(-0.644199\pi\)
							−0.298782	+	0.954321i	\(0.596580\pi\)
\(31\)	13.0972	−	8.92953i	0.422491	−	0.288049i	−0.333352	−	0.942802i	\(-0.608180\pi\)
							0.755843	+	0.654753i	\(0.227227\pi\)
\(37\)	46.9697	+	27.1180i	1.26945	+	0.732919i	0.974884	−	0.222712i	\(-0.0714909\pi\)
							0.294568	+	0.955631i	\(0.404824\pi\)
\(41\)	−4.74878	−	2.28689i	−0.115824	−	0.0557779i	0.375075	−	0.926995i	\(-0.377617\pi\)
							−0.490898	+	0.871217i	\(0.663331\pi\)
\(43\)	−37.1274	+	21.6923i	−0.863428	+	0.504472i
							\(47\)	−2.43460	−	3.05289i	−0.0518000	−	0.0649551i	0.755257	−	0.655429i	\(-0.227512\pi\)
−0.807057	+	0.590474i	\(0.798941\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.h.a.3.6	✓	72
3.2	odd	2		387.3.bn.b.46.1		72
43.29	odd	42	inner	43.3.h.a.29.6	yes	72
129.29	even	42		387.3.bn.b.244.1		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.h.a.3.6	✓	72	1.1	even	1	trivial
43.3.h.a.29.6	yes	72	43.29	odd	42	inner
387.3.bn.b.46.1		72	3.2	odd	2
387.3.bn.b.244.1		72	129.29	even	42