Embedded newform 43.3.f.a.2.1

• Label: 43.3.f.a.2.1
• Level: $43$
• Weight: $3$
• Character: 43.2
• Analytic conductor: $1.172$
• Analytic rank: $0$
• Dimension: $42$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	43.f (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.17166513675\)
Analytic rank:	\(0\)
Dimension:	\(42\)
Relative dimension:	\(7\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			2.1
Character	\(\chi\)	\(=\)	43.2
Dual form			43.3.f.a.22.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.57834 - 3.27745i) q^{2} +(-1.75180 + 3.63765i) q^{3} +(-5.75657 + 7.21851i) q^{4} +(-7.35771 - 1.67935i) q^{5} +14.6871 q^{6} -2.47266i q^{7} +(18.5581 + 4.23577i) q^{8} +(-4.55228 - 5.70838i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 7 q^{2} - 7 q^{3} + 5 q^{4} - 7 q^{5} - 20 q^{6} + 21 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{9}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.57834	−	3.27745i	−0.789168	−	1.63872i	−0.769271	−	0.638923i	\(-0.779380\pi\)
							−0.0198971	−	0.999802i	\(-0.506334\pi\)
\(3\)	−1.75180	+	3.63765i	−0.583933	+	1.21255i	0.374499	+	0.927227i	\(0.377815\pi\)
							−0.958432	+	0.285322i	\(0.907899\pi\)
\(5\)	−7.35771	−	1.67935i	−1.47154	−	0.335870i	−0.589780	−	0.807564i	\(-0.700786\pi\)
							−0.881762	+	0.471694i	\(0.843643\pi\)
\(7\)		−	2.47266i		−	0.353237i	−0.984279	−	0.176619i	\(-0.943484\pi\)
							0.984279	−	0.176619i	\(-0.0565159\pi\)
\(11\)	−9.47252	−	11.8782i	−0.861138	−	1.07983i	−0.996034	−	0.0889731i	\(-0.971641\pi\)
							0.134896	−	0.990860i	\(-0.456930\pi\)
\(13\)	−2.97513	+	13.0349i	−0.228856	+	1.00268i	0.721718	+	0.692187i	\(0.243353\pi\)
							−0.950574	+	0.310497i	\(0.899505\pi\)
\(17\)	2.29753	+	10.0661i	0.135149	+	0.592125i	0.996462	+	0.0840503i	\(0.0267856\pi\)
							−0.861313	+	0.508075i	\(0.830357\pi\)
\(19\)	−13.0078	−	10.3734i	−0.684620	−	0.545966i	0.218243	−	0.975894i	\(-0.429967\pi\)
							−0.902863	+	0.429928i	\(0.858539\pi\)
\(23\)	8.38092	+	10.5093i	0.364388	+	0.456928i	0.929900	−	0.367812i	\(-0.119893\pi\)
							−0.565512	+	0.824740i	\(0.691322\pi\)
\(29\)	−8.57324	−	17.8025i	−0.295629	−	0.613880i	0.699258	−	0.714869i	\(-0.253514\pi\)
							−0.994888	+	0.100989i	\(0.967799\pi\)
\(31\)	1.08573	−	0.522860i	0.0350235	−	0.0168664i	−0.416290	−	0.909232i	\(-0.636670\pi\)
							0.451314	+	0.892365i	\(0.350956\pi\)
\(37\)		−	13.7982i		−	0.372924i	−0.982462	−	0.186462i	\(-0.940298\pi\)
							0.982462	−	0.186462i	\(-0.0597021\pi\)
\(41\)	−45.8805	+	22.0949i	−1.11904	+	0.538899i	−0.899594	−	0.436726i	\(-0.856138\pi\)
							−0.219441	+	0.975626i	\(0.570424\pi\)
\(43\)	−37.2312	−	21.5136i	−0.865842	−	0.500317i
							\(47\)	−36.4467	+	45.7027i	−0.775462	+	0.972399i	−0.999998	−	0.00206704i	\(-0.999342\pi\)
0.224536	+	0.974466i	\(0.427913\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.f.a.2.1	✓	42
3.2	odd	2		387.3.w.b.217.7		42
43.22	odd	14	inner	43.3.f.a.22.1	yes	42
129.65	even	14		387.3.w.b.280.7		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.f.a.2.1	✓	42	1.1	even	1	trivial
43.3.f.a.22.1	yes	42	43.22	odd	14	inner
387.3.w.b.217.7		42	3.2	odd	2
387.3.w.b.280.7		42	129.65	even	14