Embedded newform 43.3.f.a.2.5

• Label: 43.3.f.a.2.5
• 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			2.5
Character	\(\chi\)	\(=\)	43.2
Dual form			43.3.f.a.22.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.311899 + 0.647664i) q^{2} +(1.16241 - 2.41377i) q^{3} +(2.17177 - 2.72331i) q^{4} +(-3.49769 - 0.798324i) q^{5} +1.92587 q^{6} +9.27538i q^{7} +(5.24449 + 1.19702i) q^{8} +(1.13631 + 1.42489i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 7 q^{2} - 7 q^{3} + 5 q^{4} - 7 q^{5} - 20 q^{6} + 21 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.311899	+	0.647664i	0.155949	+	0.323832i	0.964275	−	0.264904i	\(-0.0853401\pi\)
							−0.808326	+	0.588736i	\(0.799626\pi\)
\(3\)	1.16241	−	2.41377i	0.387471	−	0.804591i	−0.612430	−	0.790524i	\(-0.709808\pi\)
							0.999901	−	0.0140666i	\(-0.00447767\pi\)
\(5\)	−3.49769	−	0.798324i	−0.699537	−	0.159665i	−0.142061	−	0.989858i	\(-0.545373\pi\)
							−0.557476	+	0.830193i	\(0.688230\pi\)
\(7\)			9.27538i			1.32505i	0.749038	+	0.662527i	\(0.230516\pi\)
							−0.749038	+	0.662527i	\(0.769484\pi\)
\(11\)	−7.85918	−	9.85510i	−0.714471	−	0.895918i	0.283540	−	0.958960i	\(-0.408491\pi\)
							−0.998011	+	0.0630426i	\(0.979920\pi\)
\(13\)	−5.05397	+	22.1429i	−0.388767	+	1.70330i	0.280146	+	0.959957i	\(0.409617\pi\)
							−0.668912	+	0.743341i	\(0.733240\pi\)
\(17\)	−4.36571	−	19.1274i	−0.256806	−	1.12514i	−0.924643	−	0.380835i	\(-0.875637\pi\)
							0.667837	−	0.744308i	\(-0.267220\pi\)
\(19\)	2.63150	+	2.09855i	0.138500	+	0.110450i	0.690289	−	0.723534i	\(-0.257483\pi\)
							−0.551789	+	0.833984i	\(0.686055\pi\)
\(23\)	9.12272	+	11.4395i	0.396640	+	0.497371i	0.939546	−	0.342422i	\(-0.111247\pi\)
							−0.542906	+	0.839794i	\(0.682676\pi\)
\(29\)	0.220329	+	0.457519i	0.00759757	+	0.0157765i	0.904733	−	0.425979i	\(-0.140070\pi\)
							−0.897136	+	0.441755i	\(0.854356\pi\)
\(31\)	−6.91900	+	3.33201i	−0.223194	+	0.107484i	−0.542139	−	0.840289i	\(-0.682385\pi\)
							0.318946	+	0.947773i	\(0.396671\pi\)
\(37\)		−	40.3090i		−	1.08943i	−0.838620	−	0.544716i	\(-0.816637\pi\)
							0.838620	−	0.544716i	\(-0.183363\pi\)
\(41\)	−44.6617	+	21.5079i	−1.08931	+	0.524584i	−0.890282	−	0.455410i	\(-0.849493\pi\)
							−0.199028	+	0.979994i	\(0.563778\pi\)
\(43\)	42.3934	−	7.19744i	0.985892	−	0.167382i
							\(47\)	−21.6725	+	27.1764i	−0.461116	+	0.578222i	−0.956971	−	0.290184i	\(-0.906283\pi\)
0.495855	+	0.868406i	\(0.334855\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.f.a.2.5	✓	42
3.2	odd	2		387.3.w.b.217.3		42
43.22	odd	14	inner	43.3.f.a.22.5	yes	42
129.65	even	14		387.3.w.b.280.3		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.f.a.2.5	✓	42	1.1	even	1	trivial
43.3.f.a.22.5	yes	42	43.22	odd	14	inner
387.3.w.b.217.3		42	3.2	odd	2
387.3.w.b.280.3		42	129.65	even	14