Embedded newform 43.3.f.a.2.6

• Label: 43.3.f.a.2.6
• 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			2.6
Character	\(\chi\)	\(=\)	43.2
Dual form			43.3.f.a.22.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.03788 + 2.15517i) q^{2} +(-1.08703 + 2.25725i) q^{3} +(-1.07362 + 1.34627i) q^{4} +(-0.904592 - 0.206467i) q^{5} -5.99296 q^{6} -5.06547i q^{7} +(5.31261 + 1.21257i) q^{8} +(1.69788 + 2.12908i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 7 q^{2} - 7 q^{3} + 5 q^{4} - 7 q^{5} - 20 q^{6} + 21 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.03788	+	2.15517i	0.518938	+	1.07759i	0.981582	+	0.191041i	\(0.0611865\pi\)
							−0.462644	+	0.886544i	\(0.653099\pi\)
\(3\)	−1.08703	+	2.25725i	−0.362344	+	0.752416i	−0.999837	−	0.0180403i	\(-0.994257\pi\)
							0.637493	+	0.770456i	\(0.279972\pi\)
\(5\)	−0.904592	−	0.206467i	−0.180918	−	0.0412935i	0.131102	−	0.991369i	\(-0.458149\pi\)
							−0.312020	+	0.950075i	\(0.601006\pi\)
\(7\)		−	5.06547i		−	0.723639i	−0.932248	−	0.361820i	\(-0.882156\pi\)
							0.932248	−	0.361820i	\(-0.117844\pi\)
\(11\)	−1.54566	−	1.93820i	−0.140515	−	0.176200i	0.706594	−	0.707619i	\(-0.250231\pi\)
							−0.847109	+	0.531419i	\(0.821659\pi\)
\(13\)	2.68779	−	11.7760i	0.206753	−	0.905845i	−0.759957	−	0.649973i	\(-0.774780\pi\)
							0.966711	−	0.255872i	\(-0.0823627\pi\)
\(17\)	−0.329454	−	1.44343i	−0.0193796	−	0.0849077i	0.964313	−	0.264764i	\(-0.0852940\pi\)
							−0.983693	+	0.179856i	\(0.942437\pi\)
\(19\)	−17.4055	−	13.8804i	−0.916080	−	0.730550i	0.0472457	−	0.998883i	\(-0.484956\pi\)
							−0.963326	+	0.268334i	\(0.913527\pi\)
\(23\)	−5.29301	−	6.63723i	−0.230131	−	0.288575i	0.653336	−	0.757068i	\(-0.273369\pi\)
							−0.883467	+	0.468492i	\(0.844797\pi\)
\(29\)	18.1906	+	37.7731i	0.627262	+	1.30252i	0.936209	+	0.351443i	\(0.114309\pi\)
							−0.308948	+	0.951079i	\(0.599977\pi\)
\(31\)	−12.6726	+	6.10282i	−0.408794	+	0.196865i	−0.626968	−	0.779045i	\(-0.715704\pi\)
							0.218174	+	0.975910i	\(0.429990\pi\)
\(37\)			47.0512i			1.27165i	0.771831	+	0.635827i	\(0.219341\pi\)
							−0.771831	+	0.635827i	\(0.780659\pi\)
\(41\)	54.7630	−	26.3725i	1.33568	−	0.643231i	0.376605	−	0.926374i	\(-0.377091\pi\)
							0.959078	+	0.283143i	\(0.0913771\pi\)
\(43\)	−24.7769	−	35.1441i	−0.576207	−	0.817304i
							\(47\)	−14.3910	+	18.0457i	−0.306191	+	0.383952i	−0.910991	−	0.412426i	\(-0.864682\pi\)
0.604800	+	0.796377i	\(0.293253\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.f.a.2.6	✓	42
3.2	odd	2		387.3.w.b.217.2		42
43.22	odd	14	inner	43.3.f.a.22.6	yes	42
129.65	even	14		387.3.w.b.280.2		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.f.a.2.6	✓	42	1.1	even	1	trivial
43.3.f.a.22.6	yes	42	43.22	odd	14	inner
387.3.w.b.217.2		42	3.2	odd	2
387.3.w.b.280.2		42	129.65	even	14