Embedded newform 43.3.h.a.12.1

• Label: 43.3.h.a.12.1
• 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			12.1
Character	\(\chi\)	\(=\)	43.12
Dual form			43.3.h.a.18.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28258 - 2.66331i) q^{2} +(-1.59066 - 2.33308i) q^{3} +(-2.95426 + 3.70453i) q^{4} +(-1.17011 + 1.26107i) q^{5} +(-4.17355 + 7.22881i) q^{6} +(1.73703 - 1.00288i) q^{7} +(2.12764 + 0.485621i) q^{8} +(0.375040 - 0.955585i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 14 q^{2} - 14 q^{3} + 12 q^{4} - 11 q^{5} + 2 q^{6} - 30 q^{7} - 42 q^{8} + 54 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.28258	−	2.66331i	−0.641292	−	1.33166i	−0.927620	−	0.373525i	\(-0.878149\pi\)
							0.286328	−	0.958132i	\(-0.407565\pi\)
\(3\)	−1.59066	−	2.33308i	−0.530221	−	0.777692i	0.463909	−	0.885883i	\(-0.346447\pi\)
							−0.994130	+	0.108191i	\(0.965494\pi\)
\(5\)	−1.17011	+	1.26107i	−0.234021	+	0.252215i	−0.839134	−	0.543925i	\(-0.816938\pi\)
							0.605113	+	0.796140i	\(0.293128\pi\)
\(7\)	1.73703	−	1.00288i	0.248147	−	0.143268i	−0.370768	−	0.928725i	\(-0.620906\pi\)
							0.618916	+	0.785457i	\(0.287572\pi\)
\(11\)	−8.89454	−	11.1534i	−0.808594	−	1.01395i	−0.999477	−	0.0323325i	\(-0.989706\pi\)
							0.190883	−	0.981613i	\(-0.438865\pi\)
\(13\)	17.3266	−	5.34455i	1.33282	−	0.411119i	0.455160	−	0.890410i	\(-0.349582\pi\)
							0.877656	+	0.479290i	\(0.159106\pi\)
\(17\)	9.71988	−	9.01873i	0.571757	−	0.530513i	−0.340439	−	0.940267i	\(-0.610576\pi\)
							0.912196	+	0.409753i	\(0.134385\pi\)
\(19\)	−7.73572	+	3.03605i	−0.407143	+	0.159792i	−0.560069	−	0.828446i	\(-0.689226\pi\)
							0.152926	+	0.988238i	\(0.451130\pi\)
\(23\)	30.5120	−	4.59895i	1.32661	−	0.199954i	0.552769	−	0.833334i	\(-0.313571\pi\)
							0.773841	+	0.633380i	\(0.218333\pi\)
\(29\)	−17.7571	+	26.0449i	−0.612314	+	0.898100i	−0.999716	−	0.0238107i	\(-0.992420\pi\)
							0.387402	+	0.921911i	\(0.373372\pi\)
\(31\)	2.54197	−	33.9202i	0.0819990	−	1.09420i	−0.792725	−	0.609579i	\(-0.791338\pi\)
							0.874724	−	0.484621i	\(-0.161043\pi\)
\(37\)	37.2074	+	21.4817i	1.00560	+	0.580586i	0.909902	−	0.414824i	\(-0.136157\pi\)
							0.0957027	+	0.995410i	\(0.469490\pi\)
\(41\)	25.9778	−	12.5103i	0.633606	−	0.305129i	−0.0893734	−	0.995998i	\(-0.528486\pi\)
							0.722979	+	0.690870i	\(0.242772\pi\)
\(43\)	41.4963	−	11.2720i	0.965030	−	0.262139i
							\(47\)	−8.56502	+	10.7402i	−0.182235	+	0.228515i	−0.864555	−	0.502538i	\(-0.832400\pi\)
0.682320	+	0.731053i	\(0.260971\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.h.a.12.1	✓	72
3.2	odd	2		387.3.bn.b.55.6		72
43.18	odd	42	inner	43.3.h.a.18.1	yes	72
129.104	even	42		387.3.bn.b.190.6		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.h.a.12.1	✓	72	1.1	even	1	trivial
43.3.h.a.18.1	yes	72	43.18	odd	42	inner
387.3.bn.b.55.6		72	3.2	odd	2
387.3.bn.b.190.6		72	129.104	even	42