Embedded newform 43.4.g.a.10.5

• Label: 43.4.g.a.10.5
• Level: $43$
• Weight: $4$
• Character: 43.10
• 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			10.5
Character	\(\chi\)	\(=\)	43.10
Dual form			43.4.g.a.13.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.294103 - 1.28855i) q^{2} +(-1.69703 - 1.57462i) q^{3} +(5.63389 - 2.71314i) q^{4} +(1.82643 - 0.275290i) q^{5} +(-1.52987 + 2.64981i) q^{6} +(-11.2969 - 19.5668i) q^{7} +(-11.7454 - 14.7283i) q^{8} +(-1.61721 - 21.5801i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 12 q^{2} - 9 q^{3} - 92 q^{4} + 5 q^{5} - 22 q^{6} - 54 q^{7} + 2 q^{8} + 201 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.294103	−	1.28855i	−0.103981	−	0.455570i	−0.999935	−	0.0114231i	\(-0.996364\pi\)
							0.895954	−	0.444147i	\(-0.146493\pi\)
\(3\)	−1.69703	−	1.57462i	−0.326594	−	0.303035i	0.499784	−	0.866150i	\(-0.333413\pi\)
							−0.826379	+	0.563114i	\(0.809603\pi\)
\(5\)	1.82643	−	0.275290i	0.163361	−	0.0246227i	−0.0668524	−	0.997763i	\(-0.521296\pi\)
							0.230213	+	0.973140i	\(0.426058\pi\)
\(7\)	−11.2969	−	19.5668i	−0.609976	−	1.05651i	−0.991244	−	0.132045i	\(-0.957846\pi\)
							0.381267	−	0.924465i	\(-0.375488\pi\)
\(11\)	51.6605	+	24.8784i	1.41602	+	0.681920i	0.976341	−	0.216235i	\(-0.0693778\pi\)
							0.439679	+	0.898155i	\(0.355092\pi\)
\(13\)	−17.9273	+	45.6779i	−0.382471	+	0.974521i	0.601924	+	0.798553i	\(0.294401\pi\)
							−0.984396	+	0.175968i	\(0.943694\pi\)
\(17\)	39.1160	+	5.89579i	0.558061	+	0.0841141i	0.422016	−	0.906589i	\(-0.361323\pi\)
							0.136045	+	0.990703i	\(0.456561\pi\)
\(19\)	−9.44760	+	126.070i	−0.114075	+	1.52223i	0.587261	+	0.809397i	\(0.300206\pi\)
							−0.701336	+	0.712830i	\(0.747413\pi\)
\(23\)	126.479	−	86.2318i	1.14664	−	0.781764i	0.167683	−	0.985841i	\(-0.446372\pi\)
							0.978955	+	0.204077i	\(0.0654192\pi\)
\(29\)	−6.72625	+	6.24105i	−0.0430701	+	0.0399632i	−0.701414	−	0.712754i	\(-0.747448\pi\)
							0.658344	+	0.752717i	\(0.271257\pi\)
\(31\)	265.819	+	81.9944i	1.54008	+	0.475053i	0.944575	−	0.328297i	\(-0.106475\pi\)
							0.595508	+	0.803349i	\(0.296951\pi\)
\(37\)	−11.4741	+	19.8738i	−0.0509821	+	0.0883035i	−0.890390	−	0.455198i	\(-0.849568\pi\)
							0.839408	+	0.543502i	\(0.182902\pi\)
\(41\)	−58.5363	−	256.464i	−0.222972	−	0.976903i	−0.955227	−	0.295873i	\(-0.904389\pi\)
							0.732256	−	0.681030i	\(-0.238468\pi\)
\(43\)	−280.564	+	28.1244i	−0.995013	+	0.0997427i
							\(47\)	61.6290	−	29.6789i	0.191266	−	0.0921089i	−0.335800	−	0.941933i	\(-0.609006\pi\)
0.527066	+	0.849825i	\(0.323292\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.4.g.a.10.5	✓	120
43.13	even	21	inner	43.4.g.a.13.5	yes	120
43.20	odd	42		1849.4.a.k.1.26		60
43.23	even	21		1849.4.a.l.1.35		60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.g.a.10.5	✓	120	1.1	even	1	trivial
43.4.g.a.13.5	yes	120	43.13	even	21	inner
1849.4.a.k.1.26		60	43.20	odd	42
1849.4.a.l.1.35		60	43.23	even	21