Embedded newform 29.3.f.a.21.3

• Label: 29.3.f.a.21.3
• Level: $29$
• Weight: $3$
• Character: 29.21
• Analytic conductor: $0.790$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	29.f (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.790192766645\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(4\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			21.3
Character	\(\chi\)	\(=\)	29.21
Dual form			29.3.f.a.18.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.144456 + 0.412831i) q^{2} +(0.570967 + 0.0643326i) q^{3} +(2.97776 + 2.37469i) q^{4} +(-1.02777 - 2.13419i) q^{5} +(-0.109038 + 0.226420i) q^{6} +(-1.74606 - 2.18949i) q^{7} +(-2.89184 + 1.81707i) q^{8} +(-8.45249 - 1.92922i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 16 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} + 28 q^{8} - 14 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/29\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{17}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.144456	+	0.412831i	−0.0722279	+	0.206416i	−0.974338	−	0.225091i	\(-0.927732\pi\)
							0.902110	+	0.431506i	\(0.142018\pi\)
\(3\)	0.570967	+	0.0643326i	0.190322	+	0.0214442i	0.206611	−	0.978423i	\(-0.433757\pi\)
							−0.0162887	+	0.999867i	\(0.505185\pi\)
\(5\)	−1.02777	−	2.13419i	−0.205554	−	0.426837i	0.772550	−	0.634954i	\(-0.218981\pi\)
							−0.978104	+	0.208117i	\(0.933267\pi\)
\(7\)	−1.74606	−	2.18949i	−0.249438	−	0.312785i	0.641311	−	0.767281i	\(-0.278391\pi\)
							−0.890749	+	0.454496i	\(0.849819\pi\)
\(11\)	−7.71599	−	4.84828i	−0.701454	−	0.440753i	0.133517	−	0.991047i	\(-0.457373\pi\)
							−0.834971	+	0.550294i	\(0.814516\pi\)
\(13\)	8.82927	−	2.01522i	0.679175	−	0.155017i	0.131002	−	0.991382i	\(-0.458181\pi\)
							0.548173	+	0.836365i	\(0.315324\pi\)
\(17\)	5.09882	−	5.09882i	0.299931	−	0.299931i	−0.541056	−	0.840987i	\(-0.681975\pi\)
							0.840987	+	0.541056i	\(0.181975\pi\)
\(19\)	2.57525	+	22.8560i	0.135540	+	1.20295i	0.858778	+	0.512348i	\(0.171224\pi\)
							−0.723238	+	0.690599i	\(0.757347\pi\)
\(23\)	17.8554	+	8.59869i	0.776320	+	0.373856i	0.779712	−	0.626138i	\(-0.215365\pi\)
							−0.00339216	+	0.999994i	\(0.501080\pi\)
\(29\)	−26.8516	−	10.9540i	−0.925918	−	0.377724i
							\(31\)	−10.2548	+	29.3065i	−0.330800	+	0.945371i	0.651597	+	0.758565i	\(0.274099\pi\)
−0.982397	+	0.186806i	\(0.940186\pi\)
\(37\)	5.78232	−	3.63327i	0.156279	−	0.0981966i	−0.451622	−	0.892209i	\(-0.649155\pi\)
							0.607901	+	0.794013i	\(0.292012\pi\)
\(41\)	9.53079	+	9.53079i	0.232458	+	0.232458i	0.813718	−	0.581260i	\(-0.197440\pi\)
							−0.581260	+	0.813718i	\(0.697440\pi\)
\(43\)	47.8950	−	16.7592i	1.11384	−	0.389748i	0.290257	−	0.956949i	\(-0.406259\pi\)
							0.823580	+	0.567200i	\(0.191973\pi\)
\(47\)	−44.9627	+	71.5578i	−0.956654	+	1.52251i	−0.107604	+	0.994194i	\(0.534318\pi\)
							−0.849050	+	0.528312i	\(0.822825\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.3.f.a.21.3	yes	48
3.2	odd	2		261.3.s.a.253.2		48
29.18	odd	28	inner	29.3.f.a.18.3	✓	48
87.47	even	28		261.3.s.a.163.2		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.3.f.a.18.3	✓	48	29.18	odd	28	inner
29.3.f.a.21.3	yes	48	1.1	even	1	trivial
261.3.s.a.163.2		48	87.47	even	28
261.3.s.a.253.2		48	3.2	odd	2