Embedded newform 43.3.f.a.32.1

• Label: 43.3.f.a.32.1
• Level: $43$
• Weight: $3$
• Character: 43.32
• Analytic conductor: $1.172$
• Analytic rank: $0$
• Dimension: $42$
• 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			32.1
Character	\(\chi\)	\(=\)	43.32
Dual form			43.3.f.a.39.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.96269 + 2.36266i) q^{2} +(-2.18152 - 1.73970i) q^{3} +(2.30525 - 10.0999i) q^{4} +(0.910321 + 1.89030i) q^{5} +10.5735 q^{6} -8.82371i q^{7} +(10.4564 + 21.7129i) q^{8} +(-0.270235 - 1.18398i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 7 q^{2} - 7 q^{3} + 5 q^{4} - 7 q^{5} - 20 q^{6} + 21 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.96269	+	2.36266i	−1.48134	+	1.18133i	−0.541038	+	0.840998i	\(0.681969\pi\)
							−0.940305	+	0.340334i	\(0.889460\pi\)
\(3\)	−2.18152	−	1.73970i	−0.727172	−	0.579900i	0.188382	−	0.982096i	\(-0.439676\pi\)
							−0.915554	+	0.402195i	\(0.868247\pi\)
\(5\)	0.910321	+	1.89030i	0.182064	+	0.378060i	0.971948	−	0.235197i	\(-0.0755734\pi\)
							−0.789884	+	0.613257i	\(0.789859\pi\)
\(7\)		−	8.82371i		−	1.26053i	−0.776380	−	0.630265i	\(-0.782946\pi\)
							0.776380	−	0.630265i	\(-0.217054\pi\)
\(11\)	−3.64956	−	15.9898i	−0.331778	−	1.45361i	−0.815686	−	0.578495i	\(-0.803640\pi\)
							0.483908	−	0.875119i	\(-0.339217\pi\)
\(13\)	−12.2877	+	5.91743i	−0.945206	+	0.455187i	−0.842003	−	0.539473i	\(-0.818623\pi\)
							−0.103203	+	0.994660i	\(0.532909\pi\)
\(17\)	−9.25392	−	4.45645i	−0.544348	−	0.262144i	0.141438	−	0.989947i	\(-0.454827\pi\)
							−0.685786	+	0.727803i	\(0.740542\pi\)
\(19\)	22.6397	+	5.16738i	1.19157	+	0.271967i	0.771919	−	0.635721i	\(-0.219297\pi\)
							0.419646	+	0.907688i	\(0.362154\pi\)
\(23\)	2.32394	+	10.1819i	0.101041	+	0.442690i	0.999989	+	0.00463488i	\(0.00147533\pi\)
							−0.898948	+	0.438055i	\(0.855668\pi\)
\(29\)	−6.41134	+	5.11287i	−0.221081	+	0.176306i	−0.727767	−	0.685825i	\(-0.759442\pi\)
							0.506686	+	0.862131i	\(0.330870\pi\)
\(31\)	−17.8403	−	22.3710i	−0.575493	−	0.721645i	0.405844	−	0.913942i	\(-0.366978\pi\)
							−0.981337	+	0.192297i	\(0.938406\pi\)
\(37\)		−	11.3223i		−	0.306009i	−0.988226	−	0.153004i	\(-0.951105\pi\)
							0.988226	−	0.153004i	\(-0.0488949\pi\)
\(41\)	17.4372	+	21.8656i	0.425298	+	0.533307i	0.947602	−	0.319452i	\(-0.103499\pi\)
							−0.522304	+	0.852759i	\(0.674927\pi\)
\(43\)	42.1239	−	8.63601i	0.979625	−	0.200837i
							\(47\)	8.30189	−	36.3730i	0.176636	−	0.773893i	−0.806532	−	0.591190i	\(-0.798658\pi\)
0.983168	−	0.182703i	\(-0.0584846\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.f.a.32.1	✓	42
3.2	odd	2		387.3.w.b.118.7		42
43.39	odd	14	inner	43.3.f.a.39.1	yes	42
129.125	even	14		387.3.w.b.82.7		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.f.a.32.1	✓	42	1.1	even	1	trivial
43.3.f.a.39.1	yes	42	43.39	odd	14	inner
387.3.w.b.82.7		42	129.125	even	14
387.3.w.b.118.7		42	3.2	odd	2