Embedded newform 43.3.f.a.8.5

• Label: 43.3.f.a.8.5
• Level: $43$
• Weight: $3$
• Character: 43.8
• 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			8.5
Character	\(\chi\)	\(=\)	43.8
Dual form			43.3.f.a.27.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.71853 + 0.392244i) q^{2} +(0.160301 - 0.0365876i) q^{3} +(-0.804371 - 0.387365i) q^{4} +(5.40310 + 4.30882i) q^{5} +0.289833 q^{6} -6.65286i q^{7} +(-6.74303 - 5.37738i) q^{8} +(-8.08436 + 3.89322i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 7 q^{2} - 7 q^{3} + 5 q^{4} - 7 q^{5} - 20 q^{6} + 21 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{13}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.71853	+	0.392244i	0.859267	+	0.196122i	0.629382	−	0.777096i	\(-0.283308\pi\)
							0.229885	+	0.973218i	\(0.426165\pi\)
\(3\)	0.160301	−	0.0365876i	0.0534335	−	0.0121959i	−0.195721	−	0.980660i	\(-0.562705\pi\)
							0.249154	+	0.968464i	\(0.419847\pi\)
\(5\)	5.40310	+	4.30882i	1.08062	+	0.861765i	0.990953	−	0.134207i	\(-0.0428487\pi\)
							0.0896658	+	0.995972i	\(0.471420\pi\)
\(7\)		−	6.65286i		−	0.950409i	−0.879875	−	0.475205i	\(-0.842374\pi\)
							0.879875	−	0.475205i	\(-0.157626\pi\)
\(11\)	−16.0314	+	7.72034i	−1.45740	+	0.701849i	−0.983863	−	0.178923i	\(-0.942739\pi\)
							−0.473541	+	0.880772i	\(0.657024\pi\)
\(13\)	12.5671	−	15.7587i	0.966703	−	1.21221i	−0.0105093	−	0.999945i	\(-0.503345\pi\)
							0.977213	−	0.212263i	\(-0.0680833\pi\)
\(17\)	12.5452	+	15.7312i	0.737952	+	0.925363i	0.999203	−	0.0399080i	\(-0.0127065\pi\)
							−0.261251	+	0.965271i	\(0.584135\pi\)
\(19\)	−4.58474	+	9.52031i	−0.241302	+	0.501069i	−0.986086	−	0.166237i	\(-0.946838\pi\)
							0.744784	+	0.667306i	\(0.232553\pi\)
\(23\)	25.0978	−	12.0865i	1.09121	−	0.525499i	0.200325	−	0.979729i	\(-0.435800\pi\)
							0.890884	+	0.454231i	\(0.150086\pi\)
\(29\)	8.39350	+	1.91576i	0.289431	+	0.0660607i	0.364771	−	0.931097i	\(-0.381147\pi\)
							−0.0753404	+	0.997158i	\(0.524004\pi\)
\(31\)	3.83330	−	16.7948i	0.123655	−	0.541767i	−0.874712	−	0.484643i	\(-0.838950\pi\)
							0.998367	−	0.0571244i	\(-0.0181932\pi\)
\(37\)			32.9663i			0.890982i	0.895287	+	0.445491i	\(0.146971\pi\)
							−0.895287	+	0.445491i	\(0.853029\pi\)
\(41\)	−7.59767	+	33.2876i	−0.185309	+	0.811892i	0.793738	+	0.608259i	\(0.208132\pi\)
							−0.979047	+	0.203633i	\(0.934725\pi\)
\(43\)	−38.1304	−	19.8764i	−0.886754	−	0.462241i
							\(47\)	24.4816	+	11.7897i	0.520884	+	0.250845i	0.675803	−	0.737083i	\(-0.263797\pi\)
−0.154918	+	0.987927i	\(0.549511\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.f.a.8.5	✓	42
3.2	odd	2		387.3.w.b.352.3		42
43.27	odd	14	inner	43.3.f.a.27.5	yes	42
129.113	even	14		387.3.w.b.199.3		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.f.a.8.5	✓	42	1.1	even	1	trivial
43.3.f.a.27.5	yes	42	43.27	odd	14	inner
387.3.w.b.199.3		42	129.113	even	14
387.3.w.b.352.3		42	3.2	odd	2