Embedded newform 29.3.f.a.19.4

• Label: 29.3.f.a.19.4
• Level: $29$
• Weight: $3$
• Character: 29.19
• 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			19.4
Character	\(\chi\)	\(=\)	29.19
Dual form			29.3.f.a.26.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.42815 + 2.27289i) q^{2} +(0.724599 - 2.07079i) q^{3} +(-1.39088 + 2.88819i) q^{4} +(-6.69994 - 1.52922i) q^{5} +(5.74151 - 1.31046i) q^{6} +(-4.78081 + 2.30231i) q^{7} +(2.11889 - 0.238741i) q^{8} +(3.27337 + 2.61043i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 16 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} + 28 q^{8} - 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{9}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.42815	+	2.27289i	0.714076	+	1.13645i	0.984784	+	0.173784i	\(0.0555995\pi\)
							−0.270708	+	0.962662i	\(0.587258\pi\)
\(3\)	0.724599	−	2.07079i	0.241533	−	0.690262i	−0.757755	−	0.652539i	\(-0.773704\pi\)
							0.999288	−	0.0377230i	\(-0.0120105\pi\)
\(5\)	−6.69994	−	1.52922i	−1.33999	−	0.305843i	−0.508344	−	0.861154i	\(-0.669742\pi\)
							−0.831643	+	0.555311i	\(0.812599\pi\)
\(7\)	−4.78081	+	2.30231i	−0.682972	+	0.328902i	−0.742997	−	0.669295i	\(-0.766596\pi\)
							0.0600247	+	0.998197i	\(0.480882\pi\)
\(11\)	−3.10194	−	0.349505i	−0.281995	−	0.0317732i	−0.0301656	−	0.999545i	\(-0.509603\pi\)
							−0.251829	+	0.967772i	\(0.581032\pi\)
\(13\)	14.9316	−	11.9076i	1.14859	−	0.915967i	0.151220	−	0.988500i	\(-0.451680\pi\)
							0.997366	+	0.0725336i	\(0.0231085\pi\)
\(17\)	−19.1156	+	19.1156i	−1.12445	+	1.12445i	−0.133383	+	0.991065i	\(0.542584\pi\)
							−0.991065	+	0.133383i	\(0.957416\pi\)
\(19\)	3.26204	−	1.14144i	0.171686	−	0.0600756i	−0.243067	−	0.970009i	\(-0.578154\pi\)
							0.414753	+	0.909934i	\(0.363868\pi\)
\(23\)	−3.11691	−	13.6561i	−0.135518	−	0.593742i	−0.996388	−	0.0849173i	\(-0.972937\pi\)
							0.860870	−	0.508824i	\(-0.169920\pi\)
\(29\)	−22.5579	+	18.2247i	−0.777858	+	0.628440i
							\(31\)	−30.9563	−	49.2666i	−0.998590	−	1.58925i	−0.791726	−	0.610877i	\(-0.790817\pi\)
−0.206864	−	0.978370i	\(-0.566326\pi\)
\(37\)	23.4414	−	2.64121i	0.633552	−	0.0713842i	0.210653	−	0.977561i	\(-0.432441\pi\)
							0.422899	+	0.906177i	\(0.361012\pi\)
\(41\)	29.0869	+	29.0869i	0.709436	+	0.709436i	0.966417	−	0.256981i	\(-0.0827277\pi\)
							−0.256981	+	0.966417i	\(0.582728\pi\)
\(43\)	28.3964	+	17.8426i	0.660380	+	0.414945i	0.820102	−	0.572218i	\(-0.193917\pi\)
							−0.159721	+	0.987162i	\(0.551060\pi\)
\(47\)	5.42166	−	48.1186i	0.115355	−	1.02380i	−0.794040	−	0.607865i	\(-0.792026\pi\)
							0.909395	−	0.415934i	\(-0.136545\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.3.f.a.19.4	✓	48
3.2	odd	2		261.3.s.a.19.1		48
29.26	odd	28	inner	29.3.f.a.26.4	yes	48
87.26	even	28		261.3.s.a.55.1		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.3.f.a.19.4	✓	48	1.1	even	1	trivial
29.3.f.a.26.4	yes	48	29.26	odd	28	inner
261.3.s.a.19.1		48	3.2	odd	2
261.3.s.a.55.1		48	87.26	even	28