Embedded newform 43.3.f.a.22.4

• Label: 43.3.f.a.22.4
• Level: $43$
• Weight: $3$
• Character: 43.22
• 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			22.4
Character	\(\chi\)	\(=\)	43.22
Dual form			43.3.f.a.2.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.468493 + 0.972836i) q^{2} +(-1.93369 - 4.01535i) q^{3} +(1.76703 + 2.21579i) q^{4} +(8.30688 - 1.89599i) q^{5} +4.81220 q^{6} -6.84462i q^{7} +(-7.19423 + 1.64204i) q^{8} +(-6.77248 + 8.49242i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 7 q^{2} - 7 q^{3} + 5 q^{4} - 7 q^{5} - 20 q^{6} + 21 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.468493	+	0.972836i	−0.234247	+	0.486418i	−0.984645	−	0.174567i	\(-0.944147\pi\)
							0.750399	+	0.660986i	\(0.229862\pi\)
\(3\)	−1.93369	−	4.01535i	−0.644564	−	1.33845i	−0.925509	−	0.378727i	\(-0.876362\pi\)
							0.280945	−	0.959724i	\(-0.409352\pi\)
\(5\)	8.30688	−	1.89599i	1.66138	−	0.379198i	0.714207	−	0.699935i	\(-0.246788\pi\)
							0.947168	+	0.320737i	\(0.103931\pi\)
\(7\)		−	6.84462i		−	0.977803i	−0.872339	−	0.488901i	\(-0.837398\pi\)
							0.872339	−	0.488901i	\(-0.162602\pi\)
\(11\)	−8.76158	+	10.9867i	−0.796507	+	0.998789i	0.203299	+	0.979117i	\(0.434834\pi\)
							−0.999806	+	0.0196721i	\(0.993738\pi\)
\(13\)	1.36918	+	5.99876i	0.105321	+	0.461443i	0.999895	+	0.0145217i	\(0.00462258\pi\)
							−0.894573	+	0.446922i	\(0.852520\pi\)
\(17\)	−0.615116	+	2.69500i	−0.0361833	+	0.158529i	−0.989792	−	0.142520i	\(-0.954480\pi\)
							0.953609	+	0.301049i	\(0.0973368\pi\)
\(19\)	10.2883	−	8.20465i	0.541490	−	0.431824i	−0.314167	−	0.949368i	\(-0.601725\pi\)
							0.855657	+	0.517544i	\(0.173154\pi\)
\(23\)	−11.5254	+	14.4524i	−0.501106	+	0.628367i	−0.966478	−	0.256748i	\(-0.917349\pi\)
							0.465373	+	0.885115i	\(0.345920\pi\)
\(29\)	−22.3697	+	46.4512i	−0.771370	+	1.60177i	0.0270263	+	0.999635i	\(0.491396\pi\)
							−0.798397	+	0.602132i	\(0.794318\pi\)
\(31\)	−10.3497	−	4.98415i	−0.333861	−	0.160779i	0.259441	−	0.965759i	\(-0.416462\pi\)
							−0.593302	+	0.804980i	\(0.702176\pi\)
\(37\)		−	53.9211i		−	1.45733i	−0.684872	−	0.728663i	\(-0.740142\pi\)
							0.684872	−	0.728663i	\(-0.259858\pi\)
\(41\)	7.30590	+	3.51834i	0.178193	+	0.0858131i	0.520853	−	0.853646i	\(-0.325614\pi\)
							−0.342661	+	0.939459i	\(0.611328\pi\)
\(43\)	27.5919	+	32.9801i	0.641671	+	0.766980i
							\(47\)	18.4280	+	23.1080i	0.392085	+	0.491659i	0.938221	−	0.346038i	\(-0.112473\pi\)
−0.546136	+	0.837697i	\(0.683902\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.f.a.22.4	yes	42
3.2	odd	2		387.3.w.b.280.4		42
43.2	odd	14	inner	43.3.f.a.2.4	✓	42
129.2	even	14		387.3.w.b.217.4		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.f.a.2.4	✓	42	43.2	odd	14	inner
43.3.f.a.22.4	yes	42	1.1	even	1	trivial
387.3.w.b.217.4		42	129.2	even	14
387.3.w.b.280.4		42	3.2	odd	2