Embedded newform 29.3.f.a.21.4

• Label: 29.3.f.a.21.4
• 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			21.4
Character	\(\chi\)	\(=\)	29.21
Dual form			29.3.f.a.18.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.929596 - 2.65663i) q^{2} +(-1.45871 - 0.164357i) q^{3} +(-3.06622 - 2.44523i) q^{4} +(1.35861 + 2.82119i) q^{5} +(-1.79265 + 3.72247i) q^{6} +(2.26277 + 2.83742i) q^{7} +(0.186254 - 0.117031i) q^{8} +(-6.67353 - 1.52319i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 16 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} + 28 q^{8} - 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.929596	−	2.65663i	0.464798	−	1.32832i	−0.438280	−	0.898839i	\(-0.644412\pi\)
							0.903078	−	0.429478i	\(-0.141302\pi\)
\(3\)	−1.45871	−	0.164357i	−0.486237	−	0.0547857i	−0.134557	−	0.990906i	\(-0.542961\pi\)
							−0.351680	+	0.936120i	\(0.614390\pi\)
\(5\)	1.35861	+	2.82119i	0.271723	+	0.564239i	0.991521	−	0.129945i	\(-0.0414800\pi\)
							−0.719798	+	0.694183i	\(0.755766\pi\)
\(7\)	2.26277	+	2.83742i	0.323252	+	0.405346i	0.916732	−	0.399504i	\(-0.130817\pi\)
							−0.593479	+	0.804849i	\(0.702246\pi\)
\(11\)	9.83338	+	6.17872i	0.893944	+	0.561702i	0.898836	−	0.438286i	\(-0.144414\pi\)
							−0.00489180	+	0.999988i	\(0.501557\pi\)
\(13\)	−18.5672	+	4.23785i	−1.42825	+	0.325988i	−0.865611	−	0.500717i	\(-0.833070\pi\)
							−0.562636	+	0.826705i	\(0.690213\pi\)
\(17\)	−12.3318	+	12.3318i	−0.725401	+	0.725401i	−0.969700	−	0.244299i	\(-0.921442\pi\)
							0.244299	+	0.969700i	\(0.421442\pi\)
\(19\)	−2.90153	−	25.7518i	−0.152712	−	1.35536i	−0.803141	−	0.595789i	\(-0.796839\pi\)
							0.650429	−	0.759567i	\(-0.274589\pi\)
\(23\)	9.50107	+	4.57547i	0.413090	+	0.198934i	0.628874	−	0.777508i	\(-0.283516\pi\)
							−0.215784	+	0.976441i	\(0.569231\pi\)
\(29\)	14.7201	−	24.9864i	0.507590	−	0.861599i
							\(31\)	−8.22632	+	23.5095i	−0.265365	+	0.758370i	0.731687	+	0.681641i	\(0.238733\pi\)
−0.997052	+	0.0767289i	\(0.975552\pi\)
\(37\)	−25.3173	+	15.9079i	−0.684253	+	0.429944i	−0.828794	−	0.559554i	\(-0.810972\pi\)
							0.144541	+	0.989499i	\(0.453829\pi\)
\(41\)	36.7111	+	36.7111i	0.895392	+	0.895392i	0.995024	−	0.0996326i	\(-0.0317668\pi\)
							−0.0996326	+	0.995024i	\(0.531767\pi\)
\(43\)	37.8059	−	13.2289i	0.879208	−	0.307648i	0.147333	−	0.989087i	\(-0.452931\pi\)
							0.731875	+	0.681439i	\(0.238645\pi\)
\(47\)	16.7697	−	26.6888i	0.356802	−	0.567847i	−0.619302	−	0.785153i	\(-0.712584\pi\)
							0.976103	+	0.217306i	\(0.0697270\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.3.f.a.21.4	yes	48
3.2	odd	2		261.3.s.a.253.1		48
29.18	odd	28	inner	29.3.f.a.18.4	✓	48
87.47	even	28		261.3.s.a.163.1		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.3.f.a.18.4	✓	48	29.18	odd	28	inner
29.3.f.a.21.4	yes	48	1.1	even	1	trivial
261.3.s.a.163.1		48	87.47	even	28
261.3.s.a.253.1		48	3.2	odd	2