Embedded newform 29.3.f.a.26.3

• Label: 29.3.f.a.26.3
• Level: $29$
• Weight: $3$
• Character: 29.26
• 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			26.3
Character	\(\chi\)	\(=\)	29.26
Dual form			29.3.f.a.19.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.488049 - 0.776726i) q^{2} +(-1.66190 - 4.74943i) q^{3} +(1.37042 + 2.84571i) q^{4} +(-1.98497 + 0.453055i) q^{5} +(-4.50009 - 1.02712i) q^{6} +(9.56374 + 4.60566i) q^{7} +(6.52543 + 0.735239i) q^{8} +(-12.7587 + 10.1747i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 16 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} + 28 q^{8} - 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{19}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.488049	−	0.776726i	0.244025	−	0.388363i	−0.702145	−	0.712034i	\(-0.747774\pi\)
							0.946170	+	0.323671i	\(0.104917\pi\)
\(3\)	−1.66190	−	4.74943i	−0.553966	−	1.58314i	−0.791114	−	0.611669i	\(-0.790498\pi\)
							0.237148	−	0.971474i	\(-0.423787\pi\)
\(5\)	−1.98497	+	0.453055i	−0.396993	+	0.0906111i	−0.416356	−	0.909201i	\(-0.636693\pi\)
							0.0193635	+	0.999813i	\(0.493836\pi\)
\(7\)	9.56374	+	4.60566i	1.36625	+	0.657951i	0.966021	−	0.258465i	\(-0.0832165\pi\)
							0.400228	+	0.916415i	\(0.368931\pi\)
\(11\)	−11.6332	+	1.31075i	−1.05756	+	0.119159i	−0.623577	−	0.781762i	\(-0.714321\pi\)
							−0.433985	+	0.900920i	\(0.642893\pi\)
\(13\)	−11.0274	−	8.79408i	−0.848264	−	0.676468i	0.0996406	−	0.995023i	\(-0.468231\pi\)
							−0.947904	+	0.318556i	\(0.896802\pi\)
\(17\)	0.154761	+	0.154761i	0.00910357	+	0.00910357i	0.711644	−	0.702540i	\(-0.247951\pi\)
							−0.702540	+	0.711644i	\(0.747951\pi\)
\(19\)	20.3603	+	7.12436i	1.07159	+	0.374966i	0.807705	−	0.589587i	\(-0.200710\pi\)
							0.263888	+	0.964553i	\(0.414995\pi\)
\(23\)	1.51251	−	6.62676i	0.0657615	−	0.288120i	−0.931345	−	0.364138i	\(-0.881364\pi\)
							0.997106	+	0.0760183i	\(0.0242207\pi\)
\(29\)	−15.7913	−	24.3235i	−0.544528	−	0.838742i
							\(31\)	−0.406230	+	0.646512i	−0.0131042	+	0.0208552i	−0.853212	−	0.521564i	\(-0.825349\pi\)
0.840108	+	0.542419i	\(0.182492\pi\)
\(37\)	−6.96097	−	0.784313i	−0.188134	−	0.0211977i	0.0173945	−	0.999849i	\(-0.494463\pi\)
							−0.205529	+	0.978651i	\(0.565891\pi\)
\(41\)	35.8917	−	35.8917i	0.875407	−	0.875407i	−0.117648	−	0.993055i	\(-0.537536\pi\)
							0.993055	+	0.117648i	\(0.0375355\pi\)
\(43\)	18.2000	−	11.4358i	0.423256	−	0.265949i	−0.303519	−	0.952825i	\(-0.598161\pi\)
							0.726774	+	0.686876i	\(0.241019\pi\)
\(47\)	−2.57409	−	22.8457i	−0.0547678	−	0.486078i	−0.990917	−	0.134475i	\(-0.957065\pi\)
							0.936149	−	0.351603i	\(-0.114363\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.3.f.a.26.3	yes	48
3.2	odd	2		261.3.s.a.55.2		48
29.19	odd	28	inner	29.3.f.a.19.3	✓	48
87.77	even	28		261.3.s.a.19.2		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.3.f.a.19.3	✓	48	29.19	odd	28	inner
29.3.f.a.26.3	yes	48	1.1	even	1	trivial
261.3.s.a.19.2		48	87.77	even	28
261.3.s.a.55.2		48	3.2	odd	2