Embedded newform 29.3.f.a.26.4

• Label: 29.3.f.a.26.4
• 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			26.4
Character	\(\chi\)	\(=\)	29.26
Dual form			29.3.f.a.19.4

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

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.3.f.a.19.4	✓	48	29.19	odd	28	inner
29.3.f.a.26.4	yes	48	1.1	even	1	trivial
261.3.s.a.19.1		48	87.77	even	28
261.3.s.a.55.1		48	3.2	odd	2