Embedded newform 43.4.g.a.13.3

• Label: 43.4.g.a.13.3
• 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			13.3
Character	\(\chi\)	\(=\)	43.13
Dual form			43.4.g.a.10.3

$q$-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 \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.g.a.10.3	✓	120	43.10	even	21	inner
43.4.g.a.13.3	yes	120	1.1	even	1	trivial
1849.4.a.k.1.16		60	43.28	odd	42
1849.4.a.l.1.45		60	43.15	even	21