Embedded newform 29.3.f.a.15.2

• Label: 29.3.f.a.15.2
• 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			15.2
Character	\(\chi\)	\(=\)	29.15
Dual form			29.3.f.a.2.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.68783 + 0.190173i) q^{2} +(-3.78594 + 2.37886i) q^{3} +(-1.08711 + 0.248126i) q^{4} +(0.141728 + 0.113024i) q^{5} +(5.93762 - 4.73509i) q^{6} +(-1.55116 + 6.79606i) q^{7} +(8.20045 - 2.86946i) q^{8} +(4.76938 - 9.90371i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48q - 16q^{2} - 12q^{3} - 14q^{4} - 14q^{5} - 14q^{6} - 10q^{7} + 28q^{8} - 14q^{9} + O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.68783	+	0.190173i	−0.843915	+	0.0950863i	−0.523326	−	0.852132i	\(-0.675309\pi\)
							−0.320588	+	0.947219i	\(0.603881\pi\)
\(3\)	−3.78594	+	2.37886i	−1.26198	+	0.792954i	−0.985763	−	0.168142i	\(-0.946223\pi\)
							−0.276216	+	0.961096i	\(0.589080\pi\)
\(5\)	0.141728	+	0.113024i	0.0283456	+	0.0226049i	0.637560	−	0.770401i	\(-0.279944\pi\)
							−0.609215	+	0.793005i	\(0.708515\pi\)
\(7\)	−1.55116	+	6.79606i	−0.221594	+	0.970865i	0.734685	+	0.678408i	\(0.237330\pi\)
							−0.956279	+	0.292457i	\(0.905527\pi\)
\(11\)	−12.2566	−	4.28875i	−1.11423	−	0.389887i	−0.290505	−	0.956873i	\(-0.593823\pi\)
							−0.823727	+	0.566987i	\(0.808109\pi\)
\(13\)	9.66173	+	20.0628i	0.743210	+	1.54329i	0.836694	+	0.547671i	\(0.184486\pi\)
							−0.0934833	+	0.995621i	\(0.529800\pi\)
\(17\)	−5.90660	−	5.90660i	−0.347447	−	0.347447i	0.511711	−	0.859158i	\(-0.329012\pi\)
							−0.859158	+	0.511711i	\(0.829012\pi\)
\(19\)	−10.7431	+	17.0975i	−0.565425	+	0.899868i	−1.00000		0.000767373i	\(-0.999756\pi\)
							0.434575	+	0.900636i	\(0.356899\pi\)
\(23\)	15.0384	+	18.8575i	0.653843	+	0.819893i	0.992657	−	0.120960i	\(-0.0385973\pi\)
							−0.338814	+	0.940853i	\(0.610026\pi\)
\(29\)	28.5764	−	4.93832i	0.985395	−	0.170287i
							\(31\)	35.5909	−	4.01014i	1.14809	−	0.129359i	0.482655	−	0.875811i	\(-0.339673\pi\)
0.665440	+	0.746451i	\(0.268244\pi\)
\(37\)	−23.0925	+	8.08043i	−0.624123	+	0.218390i	−0.623767	−	0.781610i	\(-0.714399\pi\)
							−0.000355153		1.00000i	\(0.500113\pi\)
\(41\)	−14.1288	+	14.1288i	−0.344605	+	0.344605i	−0.858095	−	0.513490i	\(-0.828352\pi\)
							0.513490	+	0.858095i	\(0.328352\pi\)
\(43\)	−4.31425	+	38.2901i	−0.100331	+	0.890466i	0.838221	+	0.545330i	\(0.183596\pi\)
							−0.938553	+	0.345136i	\(0.887833\pi\)
\(47\)	1.53295	−	4.38091i	0.0326159	−	0.0932108i	−0.926424	−	0.376482i	\(-0.877134\pi\)
							0.959040	+	0.283271i	\(0.0914195\pi\)