Embedded newform 29.3.f.a.21.2

• Label: 29.3.f.a.21.2
• 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			21.2
Character	\(\chi\)	\(=\)	29.21
Dual form			29.3.f.a.18.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.887518 + 2.53638i) q^{2} +(-5.44603 - 0.613620i) q^{3} +(-2.51821 - 2.00821i) q^{4} +(3.04201 + 6.31680i) q^{5} +(6.38982 - 13.2686i) q^{6} +(-0.484944 - 0.608100i) q^{7} +(-1.77265 + 1.11383i) q^{8} +(20.5084 + 4.68090i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 16 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} + 28 q^{8} - 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.887518	+	2.53638i	−0.443759	+	1.26819i	0.477291	+	0.878745i	\(0.341619\pi\)
							−0.921050	+	0.389445i	\(0.872667\pi\)
\(3\)	−5.44603	−	0.613620i	−1.81534	−	0.204540i	−0.862099	−	0.506740i	\(-0.830850\pi\)
							−0.953245	+	0.302200i	\(0.902279\pi\)
\(5\)	3.04201	+	6.31680i	0.608402	+	1.26336i	0.946638	+	0.322299i	\(0.104456\pi\)
							−0.338236	+	0.941061i	\(0.609830\pi\)
\(7\)	−0.484944	−	0.608100i	−0.0692777	−	0.0868715i	0.745985	−	0.665963i	\(-0.231979\pi\)
							−0.815263	+	0.579091i	\(0.803408\pi\)
\(11\)	4.24713	+	2.66865i	0.386103	+	0.242604i	0.711071	−	0.703121i	\(-0.248211\pi\)
							−0.324968	+	0.945725i	\(0.605353\pi\)
\(13\)	−2.96174	+	0.675998i	−0.227826	+	0.0519999i	−0.334910	−	0.942250i	\(-0.608706\pi\)
							0.107084	+	0.994250i	\(0.465849\pi\)
\(17\)	5.44882	−	5.44882i	0.320519	−	0.320519i	−0.528447	−	0.848966i	\(-0.677226\pi\)
							0.848966	+	0.528447i	\(0.177226\pi\)
\(19\)	1.87808	+	16.6684i	0.0988462	+	0.877284i	0.941083	+	0.338176i	\(0.109810\pi\)
							−0.842237	+	0.539108i	\(0.818761\pi\)
\(23\)	16.4971	+	7.94460i	0.717266	+	0.345417i	0.756662	−	0.653806i	\(-0.226829\pi\)
							−0.0393955	+	0.999224i	\(0.512543\pi\)
\(29\)	10.4908	+	27.0359i	0.361753	+	0.932274i
							\(31\)	7.35600	−	21.0222i	0.237290	−	0.678137i	−0.762234	−	0.647301i	\(-0.775898\pi\)
0.999524	−	0.0308353i	\(-0.00981675\pi\)
\(37\)	6.78996	−	4.26641i	0.183512	−	0.115308i	−0.437143	−	0.899392i	\(-0.644010\pi\)
							0.620655	+	0.784084i	\(0.286867\pi\)
\(41\)	−35.8844	−	35.8844i	−0.875230	−	0.875230i	0.117807	−	0.993037i	\(-0.462414\pi\)
							−0.993037	+	0.117807i	\(0.962414\pi\)
\(43\)	61.1002	−	21.3799i	1.42093	−	0.497207i	0.493026	−	0.870014i	\(-0.335891\pi\)
							0.927909	+	0.372808i	\(0.121605\pi\)
\(47\)	−23.5776	+	37.5236i	−0.501651	+	0.798374i	−0.997546	−	0.0700088i	\(-0.977697\pi\)
							0.495895	+	0.868383i	\(0.334840\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.3.f.a.21.2	yes	48
3.2	odd	2		261.3.s.a.253.3		48
29.18	odd	28	inner	29.3.f.a.18.2	✓	48
87.47	even	28		261.3.s.a.163.3		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.3.f.a.18.2	✓	48	29.18	odd	28	inner
29.3.f.a.21.2	yes	48	1.1	even	1	trivial
261.3.s.a.163.3		48	87.47	even	28
261.3.s.a.253.3		48	3.2	odd	2