Embedded newform 29.3.f.a.2.2

• Label: 29.3.f.a.2.2
• Level: $29$
• Weight: $3$
• Character: 29.2
• 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			2.2
Character	\(\chi\)	\(=\)	29.2
Dual form			29.3.f.a.15.2

$q$-expansion

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

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{28}\right)\)

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

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.3.f.a.2.2	✓	48
3.2	odd	2		261.3.s.a.118.3		48
29.15	odd	28	inner	29.3.f.a.15.2	yes	48
87.44	even	28		261.3.s.a.73.3		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.3.f.a.2.2	✓	48	1.1	even	1	trivial
29.3.f.a.15.2	yes	48	29.15	odd	28	inner
261.3.s.a.73.3		48	87.44	even	28
261.3.s.a.118.3		48	3.2	odd	2