Embedded newform 43.3.f.a.2.2

• Label: 43.3.f.a.2.2
• 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			2.2
Character	\(\chi\)	\(=\)	43.2
Dual form			43.3.f.a.22.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.52706 - 3.17098i) q^{2} +(2.15990 - 4.48508i) q^{3} +(-5.22924 + 6.55726i) q^{4} +(5.44765 + 1.24339i) q^{5} -17.5204 q^{6} +9.10137i q^{7} +(15.0532 + 3.43580i) q^{8} +(-9.83935 - 12.3382i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 7 q^{2} - 7 q^{3} + 5 q^{4} - 7 q^{5} - 20 q^{6} + 21 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.52706	−	3.17098i	−0.763532	−	1.58549i	−0.809901	−	0.586567i	\(-0.800479\pi\)
							0.0463687	−	0.998924i	\(-0.485235\pi\)
\(3\)	2.15990	−	4.48508i	0.719967	−	1.49503i	−0.142977	−	0.989726i	\(-0.545668\pi\)
							0.862944	−	0.505300i	\(-0.168618\pi\)
\(5\)	5.44765	+	1.24339i	1.08953	+	0.248678i	0.729306	−	0.684188i	\(-0.239843\pi\)
							0.360224	+	0.932866i	\(0.382700\pi\)
\(7\)			9.10137i			1.30020i	0.759851	+	0.650098i	\(0.225272\pi\)
							−0.759851	+	0.650098i	\(0.774728\pi\)
\(11\)	−0.0631110	−	0.0791387i	−0.00573737	−	0.00719443i	0.778955	−	0.627080i	\(-0.215750\pi\)
							−0.784692	+	0.619886i	\(0.787179\pi\)
\(13\)	0.299108	−	1.31048i	0.0230083	−	0.100806i	−0.962120	−	0.272626i	\(-0.912108\pi\)
							0.985128	+	0.171820i	\(0.0549649\pi\)
\(17\)	3.77057	+	16.5199i	0.221798	+	0.971761i	0.956124	+	0.292963i	\(0.0946414\pi\)
							−0.734326	+	0.678797i	\(0.762501\pi\)
\(19\)	−19.1108	−	15.2403i	−1.00583	−	0.802123i	−0.0255382	−	0.999674i	\(-0.508130\pi\)
							−0.980293	+	0.197550i	\(0.936701\pi\)
\(23\)	−12.4034	−	15.5534i	−0.539279	−	0.676235i	0.435298	−	0.900286i	\(-0.356643\pi\)
							−0.974577	+	0.224052i	\(0.928072\pi\)
\(29\)	10.8631	+	22.5575i	0.374590	+	0.777845i	0.999997	−	0.00245422i	\(-0.000781203\pi\)
							−0.625407	+	0.780299i	\(0.715067\pi\)
\(31\)	3.71781	−	1.79040i	0.119929	−	0.0577549i	−0.372957	−	0.927849i	\(-0.621656\pi\)
							0.492886	+	0.870094i	\(0.335942\pi\)
\(37\)			35.7641i			0.966597i	0.875456	+	0.483299i	\(0.160561\pi\)
							−0.875456	+	0.483299i	\(0.839439\pi\)
\(41\)	6.53678	−	3.14795i	0.159434	−	0.0767792i	−0.352465	−	0.935825i	\(-0.614657\pi\)
							0.511899	+	0.859046i	\(0.328942\pi\)
\(43\)	−19.7420	+	38.2002i	−0.459117	+	0.888376i
							\(47\)	56.9539	−	71.4179i	1.21178	−	1.51953i	0.421454	−	0.906850i	\(-0.361520\pi\)
0.790331	−	0.612681i	\(-0.209909\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.f.a.2.2	✓	42
3.2	odd	2		387.3.w.b.217.6		42
43.22	odd	14	inner	43.3.f.a.22.2	yes	42
129.65	even	14		387.3.w.b.280.6		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.f.a.2.2	✓	42	1.1	even	1	trivial
43.3.f.a.22.2	yes	42	43.22	odd	14	inner
387.3.w.b.217.6		42	3.2	odd	2
387.3.w.b.280.6		42	129.65	even	14