Embedded newform 43.3.f.a.2.7

• Label: 43.3.f.a.2.7
• 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			2.7
Character	\(\chi\)	\(=\)	43.2
Dual form			43.3.f.a.22.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.68761 + 3.50435i) q^{2} +(2.07233 - 4.30323i) q^{3} +(-6.93850 + 8.70061i) q^{4} +(-2.02712 - 0.462677i) q^{5} +18.5773 q^{6} -2.90918i q^{7} +(-27.0314 - 6.16973i) q^{8} +(-8.61185 - 10.7989i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 7 q^{2} - 7 q^{3} + 5 q^{4} - 7 q^{5} - 20 q^{6} + 21 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.68761	+	3.50435i	0.843803	+	1.75218i	0.632175	+	0.774825i	\(0.282162\pi\)
							0.211628	+	0.977350i	\(0.432124\pi\)
\(3\)	2.07233	−	4.30323i	0.690775	−	1.43441i	−0.199921	−	0.979812i	\(-0.564069\pi\)
							0.890697	−	0.454598i	\(-0.150217\pi\)
\(5\)	−2.02712	−	0.462677i	−0.405424	−	0.0925354i	0.0149456	−	0.999888i	\(-0.495242\pi\)
							−0.420370	+	0.907353i	\(0.638100\pi\)
\(7\)		−	2.90918i		−	0.415597i	−0.978172	−	0.207799i	\(-0.933370\pi\)
							0.978172	−	0.207799i	\(-0.0666298\pi\)
\(11\)	2.92515	+	3.66802i	0.265922	+	0.333456i	0.896808	−	0.442420i	\(-0.145880\pi\)
							−0.630886	+	0.775876i	\(0.717308\pi\)
\(13\)	0.850162	−	3.72481i	0.0653971	−	0.286523i	−0.931646	−	0.363367i	\(-0.881627\pi\)
							0.997043	+	0.0768436i	\(0.0244842\pi\)
\(17\)	3.19318	+	13.9902i	0.187834	+	0.822956i	0.977756	+	0.209748i	\(0.0672642\pi\)
							−0.789921	+	0.613208i	\(0.789879\pi\)
\(19\)	9.17901	+	7.32002i	0.483106	+	0.385264i	0.834539	−	0.550949i	\(-0.185734\pi\)
							−0.351433	+	0.936213i	\(0.614306\pi\)
\(23\)	18.1215	+	22.7237i	0.787893	+	0.987986i	0.999942	+	0.0107276i	\(0.00341476\pi\)
							−0.212050	+	0.977259i	\(0.568014\pi\)
\(29\)	−10.9026	−	22.6395i	−0.375952	−	0.780673i	0.624048	−	0.781386i	\(-0.285487\pi\)
							−1.00000		0.000713701i	\(0.999773\pi\)
\(31\)	−43.5426	+	20.9690i	−1.40460	+	0.676420i	−0.974089	−	0.226166i	\(-0.927381\pi\)
							−0.430511	+	0.902585i	\(0.641667\pi\)
\(37\)		−	10.6683i		−	0.288332i	−0.989554	−	0.144166i	\(-0.953950\pi\)
							0.989554	−	0.144166i	\(-0.0460499\pi\)
\(41\)	44.4254	−	21.3941i	1.08355	−	0.521808i	0.195097	−	0.980784i	\(-0.437498\pi\)
							0.888449	+	0.458976i	\(0.151783\pi\)
\(43\)	−8.53281	−	42.1449i	−0.198438	−	0.980114i
							\(47\)	−3.61393	+	4.53173i	−0.0768922	+	0.0964198i	−0.818790	−	0.574093i	\(-0.805355\pi\)
0.741898	+	0.670513i	\(0.233926\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.f.a.2.7	✓	42
3.2	odd	2		387.3.w.b.217.1		42
43.22	odd	14	inner	43.3.f.a.22.7	yes	42
129.65	even	14		387.3.w.b.280.1		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.f.a.2.7	✓	42	1.1	even	1	trivial
43.3.f.a.22.7	yes	42	43.22	odd	14	inner
387.3.w.b.217.1		42	3.2	odd	2
387.3.w.b.280.1		42	129.65	even	14