Embedded newform 43.4.g.a.13.1

• Label: 43.4.g.a.13.1
• Level: $43$
• Weight: $4$
• Character: 43.13
• Analytic conductor: $2.537$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.53708213025\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(10\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			13.1
Character	\(\chi\)	\(=\)	43.13
Dual form			43.4.g.a.10.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.17910 + 5.16599i) q^{2} +(2.10604 - 1.95412i) q^{3} +(-18.0894 - 8.71140i) q^{4} +(-16.7285 - 2.52142i) q^{5} +(7.61172 + 13.1839i) q^{6} +(-11.8225 + 20.4772i) q^{7} +(39.9021 - 50.0356i) q^{8} +(-1.40089 + 18.6936i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 12 q^{2} - 9 q^{3} - 92 q^{4} + 5 q^{5} - 22 q^{6} - 54 q^{7} + 2 q^{8} + 201 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{16}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.17910	+	5.16599i	−0.416876	+	1.82645i	0.132886	+	0.991131i	\(0.457576\pi\)
							−0.549762	+	0.835322i	\(0.685281\pi\)
\(3\)	2.10604	−	1.95412i	0.405308	−	0.376070i	−0.451011	−	0.892518i	\(-0.648936\pi\)
							0.856319	+	0.516448i	\(0.172746\pi\)
\(5\)	−16.7285	−	2.52142i	−1.49625	−	0.225523i	−0.650663	−	0.759367i	\(-0.725509\pi\)
							−0.845583	+	0.533844i	\(0.820747\pi\)
\(7\)	−11.8225	+	20.4772i	−0.638357	+	1.10567i	0.347436	+	0.937704i	\(0.387052\pi\)
							−0.985793	+	0.167963i	\(0.946281\pi\)
\(11\)	20.8327	−	10.0325i	0.571028	−	0.274993i	−0.126007	−	0.992029i	\(-0.540216\pi\)
							0.697035	+	0.717037i	\(0.254502\pi\)
\(13\)	25.5455	+	65.0888i	0.545003	+	1.38864i	0.892536	+	0.450977i	\(0.148924\pi\)
							−0.347533	+	0.937668i	\(0.612980\pi\)
\(17\)	−25.0773	+	3.77980i	−0.357773	+	0.0539257i	−0.325470	−	0.945552i	\(-0.605522\pi\)
							−0.0323037	+	0.999478i	\(0.510284\pi\)
\(19\)	−3.27311	−	43.6766i	−0.0395212	−	0.527374i	−0.981470	−	0.191617i	\(-0.938627\pi\)
							0.941949	−	0.335757i	\(-0.108992\pi\)
\(23\)	−65.3088	−	44.5268i	−0.592079	−	0.403673i	0.229880	−	0.973219i	\(-0.426167\pi\)
							−0.821959	+	0.569546i	\(0.807119\pi\)
\(29\)	−151.515	−	140.586i	−0.970197	−	0.900211i	0.0248427	−	0.999691i	\(-0.492092\pi\)
							−0.995040	+	0.0994803i	\(0.968282\pi\)
\(31\)	−107.818	+	33.2574i	−0.624666	+	0.192684i	−0.590899	−	0.806746i	\(-0.701227\pi\)
							−0.0337674	+	0.999430i	\(0.510751\pi\)
\(37\)	98.4071	+	170.446i	0.437244	+	0.757329i	0.997476	−	0.0710072i	\(-0.0226213\pi\)
							−0.560232	+	0.828336i	\(0.689288\pi\)
\(41\)	−26.6511	+	116.766i	−0.101517	+	0.444775i	0.898467	+	0.439042i	\(0.144682\pi\)
							−0.999984	+	0.00573298i	\(0.998175\pi\)
\(43\)	150.786	+	238.266i	0.534758	+	0.845005i
							\(47\)	141.399	+	68.0941i	0.438833	+	0.211331i	0.640237	−	0.768177i	\(-0.278836\pi\)
−0.201404	+	0.979508i	\(0.564550\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.4.g.a.13.1	yes	120
43.10	even	21	inner	43.4.g.a.10.1	✓	120
43.15	even	21		1849.4.a.l.1.58		60
43.28	odd	42		1849.4.a.k.1.3		60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.g.a.10.1	✓	120	43.10	even	21	inner
43.4.g.a.13.1	yes	120	1.1	even	1	trivial
1849.4.a.k.1.3		60	43.28	odd	42
1849.4.a.l.1.58		60	43.15	even	21