Embedded newform 43.4.g.a.10.3

• Label: 43.4.g.a.10.3
• 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			10.3
Character	\(\chi\)	\(=\)	43.10
Dual form			43.4.g.a.13.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.757202 - 3.31752i) q^{2} +(4.24926 + 3.94274i) q^{3} +(-3.22482 + 1.55299i) q^{4} +(10.1014 - 1.52254i) q^{5} +(9.86256 - 17.0825i) q^{6} +(4.90816 + 8.50119i) q^{7} +(-9.37914 - 11.7611i) q^{8} +(0.493326 + 6.58297i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 12 q^{2} - 9 q^{3} - 92 q^{4} + 5 q^{5} - 22 q^{6} - 54 q^{7} + 2 q^{8} + 201 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.757202	−	3.31752i	−0.267711	−	1.17292i	−0.912668	−	0.408701i	\(-0.865982\pi\)
							0.644957	−	0.764219i	\(-0.276875\pi\)
\(3\)	4.24926	+	3.94274i	0.817771	+	0.758781i	0.973419	−	0.229032i	\(-0.0735560\pi\)
							−0.155648	+	0.987813i	\(0.549747\pi\)
\(5\)	10.1014	−	1.52254i	0.903494	−	0.136180i	0.319167	−	0.947698i	\(-0.396597\pi\)
							0.584326	+	0.811519i	\(0.301359\pi\)
\(7\)	4.90816	+	8.50119i	0.265016	+	0.459021i	0.967568	−	0.252611i	\(-0.0812894\pi\)
							−0.702552	+	0.711633i	\(0.747956\pi\)
\(11\)	−25.4872	−	12.2740i	−0.698607	−	0.336431i	0.0506447	−	0.998717i	\(-0.483872\pi\)
							−0.749252	+	0.662285i	\(0.769587\pi\)
\(13\)	−2.78501	+	7.09610i	−0.0594172	+	0.151393i	−0.957500	−	0.288435i	\(-0.906865\pi\)
							0.898082	+	0.439827i	\(0.144960\pi\)
\(17\)	50.9616	+	7.68122i	0.727058	+	0.109586i	0.502134	−	0.864790i	\(-0.332548\pi\)
							0.224924	+	0.974376i	\(0.427787\pi\)
\(19\)	−6.34835	+	84.7128i	−0.0766532	+	1.02287i	0.817741	+	0.575586i	\(0.195226\pi\)
							−0.894394	+	0.447279i	\(0.852393\pi\)
\(23\)	−106.823	+	72.8306i	−0.968439	+	0.660271i	−0.940663	−	0.339343i	\(-0.889795\pi\)
							−0.0277767	+	0.999614i	\(0.508843\pi\)
\(29\)	79.6248	−	73.8810i	0.509860	−	0.473081i	−0.382787	−	0.923837i	\(-0.625036\pi\)
							0.892648	+	0.450755i	\(0.148845\pi\)
\(31\)	−297.841	−	91.8717i	−1.72561	−	0.532279i	−0.736113	−	0.676858i	\(-0.763341\pi\)
							−0.989493	+	0.144579i	\(0.953817\pi\)
\(37\)	−14.2897	+	24.7505i	−0.0634923	+	0.109972i	−0.896024	−	0.444005i	\(-0.853557\pi\)
							0.832532	+	0.553977i	\(0.186891\pi\)
\(41\)	70.8275	+	310.316i	0.269790	+	1.18203i	0.910258	+	0.414042i	\(0.135884\pi\)
							−0.640467	+	0.767985i	\(0.721259\pi\)
\(43\)	−212.997	+	184.768i	−0.755390	+	0.655276i
							\(47\)	366.697	−	176.592i	1.13805	−	0.548055i	0.232625	−	0.972567i	\(-0.425269\pi\)
0.905422	+	0.424512i	\(0.139554\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.4.g.a.10.3	✓	120
43.13	even	21	inner	43.4.g.a.13.3	yes	120
43.20	odd	42		1849.4.a.k.1.16		60
43.23	even	21		1849.4.a.l.1.45		60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.g.a.10.3	✓	120	1.1	even	1	trivial
43.4.g.a.13.3	yes	120	43.13	even	21	inner
1849.4.a.k.1.16		60	43.20	odd	42
1849.4.a.l.1.45		60	43.23	even	21