Embedded newform 29.3.f.a.18.4

• Label: 29.3.f.a.18.4
• Level: $29$
• Weight: $3$
• Character: 29.18
• 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			18.4
Character	\(\chi\)	\(=\)	29.18
Dual form			29.3.f.a.21.4

$q$-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 \(\left(\mathbb{Z}/29\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{11}{28}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.3.f.a.18.4	✓	48	1.1	even	1	trivial
29.3.f.a.21.4	yes	48	29.21	odd	28	inner
261.3.s.a.163.1		48	3.2	odd	2
261.3.s.a.253.1		48	87.50	even	28