Embedded newform 43.3.h.a.12.6

• Label: 43.3.h.a.12.6
• 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			12.6
Character	\(\chi\)	\(=\)	43.12
Dual form			43.3.h.a.18.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.47143 + 3.05545i) q^{2} +(-0.584290 - 0.856996i) q^{3} +(-4.67672 + 5.86443i) q^{4} +(-0.262700 + 0.283123i) q^{5} +(1.75877 - 3.04628i) q^{6} +(4.87516 - 2.81467i) q^{7} +(-11.5749 - 2.64189i) q^{8} +(2.89502 - 7.37640i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 14 q^{2} - 14 q^{3} + 12 q^{4} - 11 q^{5} + 2 q^{6} - 30 q^{7} - 42 q^{8} + 54 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.47143	+	3.05545i	0.735714	+	1.52773i	0.845621	+	0.533784i	\(0.179230\pi\)
							−0.109907	+	0.993942i	\(0.535055\pi\)
\(3\)	−0.584290	−	0.856996i	−0.194763	−	0.285665i	0.716418	−	0.697672i	\(-0.245781\pi\)
							−0.911181	+	0.412006i	\(0.864828\pi\)
\(5\)	−0.262700	+	0.283123i	−0.0525400	+	0.0566246i	−0.758775	−	0.651353i	\(-0.774202\pi\)
							0.706235	+	0.707978i	\(0.250392\pi\)
\(7\)	4.87516	−	2.81467i	0.696451	−	0.402096i	−0.109573	−	0.993979i	\(-0.534948\pi\)
							0.806024	+	0.591882i	\(0.201615\pi\)
\(11\)	−0.00121352	−	0.00152170i	−0.000110320	−	0.000138336i	0.781776	−	0.623559i	\(-0.214314\pi\)
							−0.781887	+	0.623421i	\(0.785742\pi\)
\(13\)	−15.9165	+	4.90960i	−1.22435	+	0.377661i	−0.838500	−	0.544901i	\(-0.816567\pi\)
							−0.385848	+	0.922563i	\(0.626091\pi\)
\(17\)	23.3264	−	21.6438i	1.37214	−	1.27316i	0.446490	−	0.894789i	\(-0.352674\pi\)
							0.925653	−	0.378374i	\(-0.123517\pi\)
\(19\)	−18.8761	+	7.40831i	−0.993477	+	0.389911i	−0.805715	−	0.592304i	\(-0.798219\pi\)
							−0.187762	+	0.982215i	\(0.560123\pi\)
\(23\)	−31.7017	+	4.77826i	−1.37833	+	0.207750i	−0.796048	−	0.605234i	\(-0.793080\pi\)
							−0.582286	+	0.812984i	\(0.697842\pi\)
\(29\)	2.19893	−	3.22524i	0.0758252	−	0.111215i	−0.786453	−	0.617650i	\(-0.788085\pi\)
							0.862278	+	0.506435i	\(0.169037\pi\)
\(31\)	1.68229	−	22.4486i	0.0542674	−	0.724148i	−0.902015	−	0.431704i	\(-0.857913\pi\)
							0.956283	−	0.292444i	\(-0.0944684\pi\)
\(37\)	0.370347	+	0.213820i	0.0100094	+	0.00577891i	0.504996	−	0.863121i	\(-0.331494\pi\)
							−0.494987	+	0.868900i	\(0.664827\pi\)
\(41\)	44.3777	−	21.3712i	1.08238	−	0.521249i	0.194306	−	0.980941i	\(-0.437755\pi\)
							0.888078	+	0.459692i	\(0.152040\pi\)
\(43\)	10.6659	+	41.6562i	0.248045	+	0.968749i
							\(47\)	0.177167	−	0.222160i	0.00376950	−	0.00472681i	−0.779943	−	0.625850i	\(-0.784752\pi\)
0.783713	+	0.621124i	\(0.213323\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.h.a.12.6	✓	72
3.2	odd	2		387.3.bn.b.55.1		72
43.18	odd	42	inner	43.3.h.a.18.6	yes	72
129.104	even	42		387.3.bn.b.190.1		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.h.a.12.6	✓	72	1.1	even	1	trivial
43.3.h.a.18.6	yes	72	43.18	odd	42	inner
387.3.bn.b.55.1		72	3.2	odd	2
387.3.bn.b.190.1		72	129.104	even	42