Embedded newform 43.3.f.a.27.3

• Label: 43.3.f.a.27.3
• Level: $43$
• Weight: $3$
• Character: 43.27
• 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			27.3
Character	\(\chi\)	\(=\)	43.27
Dual form			43.3.f.a.8.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.844666 + 0.192789i) q^{2} +(4.86248 + 1.10983i) q^{3} +(-2.92758 + 1.40985i) q^{4} +(0.831553 - 0.663142i) q^{5} -4.32114 q^{6} +0.302383i q^{7} +(4.91050 - 3.91600i) q^{8} +(14.3033 + 6.88811i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 7 q^{2} - 7 q^{3} + 5 q^{4} - 7 q^{5} - 20 q^{6} + 21 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.844666	+	0.192789i	−0.422333	+	0.0963947i	−0.428405	−	0.903587i	\(-0.640924\pi\)
							0.00607209	+	0.999982i	\(0.498067\pi\)
\(3\)	4.86248	+	1.10983i	1.62083	+	0.369943i	0.934111	−	0.356982i	\(-0.116194\pi\)
							0.686717	+	0.726925i	\(0.259051\pi\)
\(5\)	0.831553	−	0.663142i	0.166311	−	0.132628i	−0.536798	−	0.843711i	\(-0.680366\pi\)
							0.703109	+	0.711083i	\(0.251795\pi\)
\(7\)			0.302383i			0.0431975i	0.999767	+	0.0215988i	\(0.00687564\pi\)
							−0.999767	+	0.0215988i	\(0.993124\pi\)
\(11\)	−4.95809	−	2.38769i	−0.450735	−	0.217063i	0.194723	−	0.980858i	\(-0.437619\pi\)
							−0.645458	+	0.763796i	\(0.723333\pi\)
\(13\)	−12.6441	−	15.8552i	−0.972623	−	1.21963i	−0.975582	−	0.219635i	\(-0.929513\pi\)
							0.00295926	−	0.999996i	\(-0.499058\pi\)
\(17\)	−13.9269	+	17.4637i	−0.819227	+	1.02728i	0.179823	+	0.983699i	\(0.442448\pi\)
							−0.999050	+	0.0435792i	\(0.986124\pi\)
\(19\)	−11.5294	−	23.9411i	−0.606813	−	1.26006i	−0.947462	−	0.319870i	\(-0.896361\pi\)
							0.340649	−	0.940191i	\(-0.389353\pi\)
\(23\)	25.4900	+	12.2753i	1.10826	+	0.533711i	0.896247	−	0.443556i	\(-0.146283\pi\)
							0.212015	+	0.977266i	\(0.431997\pi\)
\(29\)	32.2996	−	7.37217i	1.11378	−	0.254213i	0.374244	−	0.927330i	\(-0.377902\pi\)
							0.739535	+	0.673118i	\(0.235045\pi\)
\(31\)	1.10365	+	4.83540i	0.0356016	+	0.155981i	0.989604	−	0.143817i	\(-0.0459378\pi\)
							−0.954003	+	0.299798i	\(0.903081\pi\)
\(37\)		−	24.6655i		−	0.666635i	−0.942815	−	0.333318i	\(-0.891832\pi\)
							0.942815	−	0.333318i	\(-0.108168\pi\)
\(41\)	12.5719	+	55.0811i	0.306632	+	1.34344i	0.859910	+	0.510445i	\(0.170519\pi\)
							−0.553279	+	0.832996i	\(0.686624\pi\)
\(43\)	−16.1683	+	39.8445i	−0.376008	+	0.926616i
							\(47\)	−11.2262	+	5.40625i	−0.238855	+	0.115027i	−0.549483	−	0.835505i	\(-0.685175\pi\)
0.310627	+	0.950532i	\(0.399461\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.f.a.27.3	yes	42
3.2	odd	2		387.3.w.b.199.5		42
43.8	odd	14	inner	43.3.f.a.8.3	✓	42
129.8	even	14		387.3.w.b.352.5		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.f.a.8.3	✓	42	43.8	odd	14	inner
43.3.f.a.27.3	yes	42	1.1	even	1	trivial
387.3.w.b.199.5		42	3.2	odd	2
387.3.w.b.352.5		42	129.8	even	14