Embedded newform 43.4.g.a.9.3

• Label: 43.4.g.a.9.3
• Level: $43$
• Weight: $4$
• Character: 43.9
• 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			9.3
Character	\(\chi\)	\(=\)	43.9
Dual form			43.4.g.a.24.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.84616 - 2.31501i) q^{2} +(0.851997 + 0.128418i) q^{3} +(-0.170806 + 0.748350i) q^{4} +(-8.91021 - 6.07488i) q^{5} +(-1.27563 - 2.20946i) q^{6} +(4.30159 - 7.45057i) q^{7} +(-19.2945 + 9.29173i) q^{8} +(-25.0911 - 7.73957i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 12 q^{2} - 9 q^{3} - 92 q^{4} + 5 q^{5} - 22 q^{6} - 54 q^{7} + 2 q^{8} + 201 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.84616	−	2.31501i	−0.652717	−	0.818481i	0.339812	−	0.940493i	\(-0.389637\pi\)
							−0.992529	+	0.122013i	\(0.961065\pi\)
\(3\)	0.851997	+	0.128418i	0.163967	+	0.0247140i	0.230513	−	0.973069i	\(-0.425960\pi\)
							−0.0665457	+	0.997783i	\(0.521198\pi\)
\(5\)	−8.91021	−	6.07488i	−0.796954	−	0.543354i	0.0950017	−	0.995477i	\(-0.469714\pi\)
							−0.891956	+	0.452123i	\(0.850667\pi\)
\(7\)	4.30159	−	7.45057i	0.232264	−	0.402293i	−0.726210	−	0.687473i	\(-0.758720\pi\)
							0.958474	+	0.285180i	\(0.0920533\pi\)
\(11\)	−0.134045	−	0.587291i	−0.00367420	−	0.0160977i	0.973058	−	0.230562i	\(-0.0740565\pi\)
							−0.976732	+	0.214464i	\(0.931199\pi\)
\(13\)	2.09097	−	27.9021i	0.0446101	−	0.595281i	−0.929444	−	0.368964i	\(-0.879713\pi\)
							0.974054	−	0.226317i	\(-0.0726684\pi\)
\(17\)	55.0315	−	37.5198i	0.785123	−	0.535288i	−0.103116	−	0.994669i	\(-0.532881\pi\)
							0.888239	+	0.459382i	\(0.151929\pi\)
\(19\)	154.353	−	47.6114i	1.86373	−	0.574885i	0.866617	−	0.498975i	\(-0.166290\pi\)
							0.997115	−	0.0759104i	\(-0.0241863\pi\)
\(23\)	−33.1040	+	30.7160i	−0.300116	+	0.278467i	−0.815827	−	0.578296i	\(-0.803718\pi\)
							0.515712	+	0.856762i	\(0.327528\pi\)
\(29\)	−46.8714	+	7.06473i	−0.300131	+	0.0452375i	−0.297381	−	0.954759i	\(-0.596113\pi\)
							−0.00274959	+	0.999996i	\(0.500875\pi\)
\(31\)	−4.41961	+	11.2610i	−0.0256060	+	0.0652430i	−0.943105	−	0.332495i	\(-0.892109\pi\)
							0.917499	+	0.397738i	\(0.130205\pi\)
\(37\)	94.0547	+	162.907i	0.417905	+	0.723833i	0.995729	−	0.0923288i	\(-0.0294311\pi\)
							−0.577823	+	0.816162i	\(0.696098\pi\)
\(41\)	−40.0563	−	50.2290i	−0.152579	−	0.191328i	0.699667	−	0.714469i	\(-0.253332\pi\)
							−0.852246	+	0.523141i	\(0.824760\pi\)
\(43\)	−118.858	−	255.695i	−0.421527	−	0.906816i
							\(47\)	46.8246	−	205.152i	0.145321	−	0.636691i	−0.848828	−	0.528669i	\(-0.822691\pi\)
0.994149	−	0.108022i	\(-0.0344517\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.4.g.a.9.3	✓	120
43.14	even	21		1849.4.a.l.1.15		60
43.24	even	21	inner	43.4.g.a.24.3	yes	120
43.29	odd	42		1849.4.a.k.1.46		60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.g.a.9.3	✓	120	1.1	even	1	trivial
43.4.g.a.24.3	yes	120	43.24	even	21	inner
1849.4.a.k.1.46		60	43.29	odd	42
1849.4.a.l.1.15		60	43.14	even	21