Embedded newform 43.3.f.a.8.6

• Label: 43.3.f.a.8.6
• 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			8.6
Character	\(\chi\)	\(=\)	43.8
Dual form			43.3.f.a.27.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.11328 + 0.482341i) q^{2} +(1.61544 - 0.368714i) q^{3} +(0.629407 + 0.303106i) q^{4} +(-3.78878 - 3.02145i) q^{5} +3.59172 q^{6} +10.9446i q^{7} +(-5.59495 - 4.46183i) q^{8} +(-5.63502 + 2.71368i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 7 q^{2} - 7 q^{3} + 5 q^{4} - 7 q^{5} - 20 q^{6} + 21 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.11328	+	0.482341i	1.05664	+	0.241171i	0.715334	−	0.698782i	\(-0.246274\pi\)
							0.341304	+	0.939953i	\(0.389132\pi\)
\(3\)	1.61544	−	0.368714i	0.538481	−	0.122905i	0.0553732	−	0.998466i	\(-0.482365\pi\)
							0.483108	+	0.875561i	\(0.339508\pi\)
\(5\)	−3.78878	−	3.02145i	−0.757755	−	0.604289i	0.166511	−	0.986040i	\(-0.446750\pi\)
							−0.924266	+	0.381750i	\(0.875321\pi\)
\(7\)			10.9446i			1.56351i	0.623585	+	0.781756i	\(0.285676\pi\)
							−0.623585	+	0.781756i	\(0.714324\pi\)
\(11\)	16.6935	−	8.03916i	1.51759	−	0.730832i	0.524858	−	0.851190i	\(-0.324118\pi\)
							0.992731	+	0.120357i	\(0.0384040\pi\)
\(13\)	−0.601017	+	0.753651i	−0.0462320	+	0.0579732i	−0.804410	−	0.594075i	\(-0.797518\pi\)
							0.758178	+	0.652048i	\(0.226090\pi\)
\(17\)	1.47234	+	1.84626i	0.0866084	+	0.108603i	0.823248	−	0.567682i	\(-0.192160\pi\)
							−0.736639	+	0.676286i	\(0.763588\pi\)
\(19\)	7.29294	−	15.1440i	0.383839	−	0.797050i	−0.616117	−	0.787655i	\(-0.711295\pi\)
							0.999956	−	0.00939532i	\(-0.00299067\pi\)
\(23\)	−9.56855	+	4.60797i	−0.416024	+	0.200346i	−0.630173	−	0.776454i	\(-0.717016\pi\)
							0.214150	+	0.976801i	\(0.431302\pi\)
\(29\)	50.5444	+	11.5364i	1.74291	+	0.397808i	0.971219	−	0.238189i	\(-0.0765537\pi\)
							0.771692	+	0.635997i	\(0.219411\pi\)
\(31\)	−6.77442	+	29.6807i	−0.218530	+	0.957442i	0.740035	+	0.672568i	\(0.234809\pi\)
							−0.958565	+	0.284874i	\(0.908048\pi\)
\(37\)		−	0.670541i		−	0.0181227i	−0.999959	−	0.00906137i	\(-0.997116\pi\)
							0.999959	−	0.00906137i	\(-0.00288436\pi\)
\(41\)	−4.07546	+	17.8558i	−0.0994015	+	0.435506i	0.900598	+	0.434653i	\(0.143129\pi\)
							−1.00000		0.000853686i	\(0.999728\pi\)
\(43\)	−35.7189	−	23.9407i	−0.830673	−	0.556761i
							\(47\)	−46.4590	−	22.3735i	−0.988489	−	0.476031i	−0.131472	−	0.991320i	\(-0.541970\pi\)
−0.857017	+	0.515289i	\(0.827685\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.f.a.8.6	✓	42
3.2	odd	2		387.3.w.b.352.2		42
43.27	odd	14	inner	43.3.f.a.27.6	yes	42
129.113	even	14		387.3.w.b.199.2		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.f.a.8.6	✓	42	1.1	even	1	trivial
43.3.f.a.27.6	yes	42	43.27	odd	14	inner
387.3.w.b.199.2		42	129.113	even	14
387.3.w.b.352.2		42	3.2	odd	2