Embedded newform 43.4.g.a.10.8

• Label: 43.4.g.a.10.8
• 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			10.8
Character	\(\chi\)	\(=\)	43.10
Dual form			43.4.g.a.13.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.747545 + 3.27521i) q^{2} +(-6.89751 - 6.39996i) q^{3} +(-2.96041 + 1.42566i) q^{4} +(-13.7333 + 2.06996i) q^{5} +(15.8050 - 27.3750i) q^{6} +(-15.9110 - 27.5586i) q^{7} +(9.87423 + 12.3819i) q^{8} +(4.59853 + 61.3631i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 12 q^{2} - 9 q^{3} - 92 q^{4} + 5 q^{5} - 22 q^{6} - 54 q^{7} + 2 q^{8} + 201 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.747545	+	3.27521i	0.264297	+	1.15796i	0.916538	+	0.399949i	\(0.130972\pi\)
							−0.652241	+	0.758012i	\(0.726171\pi\)
\(3\)	−6.89751	−	6.39996i	−1.32743	−	1.23167i	−0.952471	−	0.304630i	\(-0.901467\pi\)
							−0.374956	−	0.927042i	\(-0.622342\pi\)
\(5\)	−13.7333	+	2.06996i	−1.22834	+	0.185143i	−0.730979	−	0.682400i	\(-0.760936\pi\)
							−0.497363	+	0.867543i	\(0.665698\pi\)
\(7\)	−15.9110	−	27.5586i	−0.859111	−	1.48802i	−0.872778	−	0.488118i	\(-0.837684\pi\)
							0.0136666	−	0.999907i	\(-0.495650\pi\)
\(11\)	4.43182	+	2.13425i	0.121477	+	0.0585002i	0.493635	−	0.869669i	\(-0.335668\pi\)
							−0.372158	+	0.928169i	\(0.621382\pi\)
\(13\)	4.71211	−	12.0063i	0.100531	−	0.256149i	−0.871755	−	0.489941i	\(-0.837018\pi\)
							0.972287	+	0.233792i	\(0.0751135\pi\)
\(17\)	−86.4223	−	13.0261i	−1.23297	−	0.185840i	−0.499954	−	0.866052i	\(-0.666650\pi\)
							−0.733016	+	0.680211i	\(0.761888\pi\)
\(19\)	5.50353	−	73.4395i	0.0664525	−	0.886747i	−0.859990	−	0.510311i	\(-0.829530\pi\)
							0.926443	−	0.376436i	\(-0.122851\pi\)
\(23\)	15.6445	−	10.6662i	0.141830	−	0.0966984i	−0.490326	−	0.871539i	\(-0.663122\pi\)
							0.632157	+	0.774841i	\(0.282170\pi\)
\(29\)	126.389	−	117.272i	0.809306	−	0.750926i	−0.162500	−	0.986709i	\(-0.551956\pi\)
							0.971806	+	0.235782i	\(0.0757652\pi\)
\(31\)	−21.4337	−	6.61141i	−0.124181	−	0.0383046i	0.232042	−	0.972706i	\(-0.425459\pi\)
							−0.356223	+	0.934401i	\(0.615936\pi\)
\(37\)	93.5004	−	161.947i	0.415443	−	0.719568i	−0.580032	−	0.814594i	\(-0.696960\pi\)
							0.995475	+	0.0950258i	\(0.0302934\pi\)
\(41\)	23.3344	+	102.235i	0.0888835	+	0.389424i	0.999728	−	0.0233258i	\(-0.00742550\pi\)
							−0.910844	+	0.412750i	\(0.864568\pi\)
\(43\)	−265.224	−	95.7258i	−0.940610	−	0.339489i
							\(47\)	−190.069	+	91.5326i	−0.589882	+	0.284072i	−0.704918	−	0.709289i	\(-0.749016\pi\)
0.115036	+	0.993361i	\(0.463302\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.4.g.a.10.8	✓	120
43.13	even	21	inner	43.4.g.a.13.8	yes	120
43.20	odd	42		1849.4.a.k.1.48		60
43.23	even	21		1849.4.a.l.1.13		60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.g.a.10.8	✓	120	1.1	even	1	trivial
43.4.g.a.13.8	yes	120	43.13	even	21	inner
1849.4.a.k.1.48		60	43.20	odd	42
1849.4.a.l.1.13		60	43.23	even	21