Embedded newform 29.3.f.a.15.3

• Label: 29.3.f.a.15.3
• Level: $29$
• Weight: $3$
• Character: 29.15
• 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			15.3
Character	\(\chi\)	\(=\)	29.15
Dual form			29.3.f.a.2.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.415096 - 0.0467701i) q^{2} +(2.68233 - 1.68542i) q^{3} +(-3.72959 + 0.851255i) q^{4} +(0.738700 + 0.589093i) q^{5} +(1.03460 - 0.825065i) q^{6} +(-0.577468 + 2.53005i) q^{7} +(-3.08546 + 1.07965i) q^{8} +(0.449302 - 0.932984i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48q - 16q^{2} - 12q^{3} - 14q^{4} - 14q^{5} - 14q^{6} - 10q^{7} + 28q^{8} - 14q^{9} + O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{27}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.415096	−	0.0467701i	0.207548	−	0.0233851i	−0.00757815	−	0.999971i	\(-0.502412\pi\)
							0.215126	+	0.976586i	\(0.430984\pi\)
\(3\)	2.68233	−	1.68542i	0.894110	−	0.561807i	−0.00477622	−	0.999989i	\(-0.501520\pi\)
							0.898886	+	0.438182i	\(0.144377\pi\)
\(5\)	0.738700	+	0.589093i	0.147740	+	0.117819i	0.694571	−	0.719425i	\(-0.255594\pi\)
							−0.546831	+	0.837243i	\(0.684166\pi\)
\(7\)	−0.577468	+	2.53005i	−0.0824954	+	0.361436i	−0.999280	−	0.0379466i	\(-0.987918\pi\)
							0.916784	+	0.399383i	\(0.130775\pi\)
\(11\)	−8.65125	−	3.02720i	−0.786477	−	0.275200i	−0.0929879	−	0.995667i	\(-0.529642\pi\)
							−0.693489	+	0.720467i	\(0.743928\pi\)
\(13\)	−4.51239	−	9.37008i	−0.347107	−	0.720775i	0.652198	−	0.758049i	\(-0.273847\pi\)
							−0.999305	+	0.0372733i	\(0.988133\pi\)
\(17\)	21.4949	+	21.4949i	1.26441	+	1.26441i	0.948936	+	0.315469i	\(0.102162\pi\)
							0.315469	+	0.948936i	\(0.397838\pi\)
\(19\)	14.2589	−	22.6929i	0.750467	−	1.19436i	−0.225030	−	0.974352i	\(-0.572248\pi\)
							0.975498	−	0.220010i	\(-0.0706090\pi\)
\(23\)	−2.27710	−	2.85539i	−0.0990042	−	0.124147i	0.729861	−	0.683595i	\(-0.239585\pi\)
							−0.828866	+	0.559448i	\(0.811013\pi\)
\(29\)	−24.3191	+	15.7981i	−0.838591	+	0.544761i
							\(31\)	21.7090	−	2.44602i	0.700291	−	0.0789039i	0.245362	−	0.969431i	\(-0.421093\pi\)
0.454929	+	0.890528i	\(0.349665\pi\)
\(37\)	−46.0478	+	16.1128i	−1.24454	+	0.435482i	−0.870608	−	0.491976i	\(-0.836275\pi\)
							−0.373927	+	0.927458i	\(0.621989\pi\)
\(41\)	−25.3355	+	25.3355i	−0.617940	+	0.617940i	−0.945003	−	0.327063i	\(-0.893941\pi\)
							0.327063	+	0.945003i	\(0.393941\pi\)
\(43\)	2.78030	−	24.6759i	0.0646582	−	0.573857i	−0.919057	−	0.394124i	\(-0.871048\pi\)
							0.983715	−	0.179733i	\(-0.0575234\pi\)
\(47\)	7.49184	−	21.4104i	0.159401	−	0.455541i	−0.836289	−	0.548289i	\(-0.815279\pi\)
							0.995690	+	0.0927479i	\(0.0295651\pi\)