Embedded newform 29.3.f.a.2.4

• Label: 29.3.f.a.2.4
• 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			2.4
Character	\(\chi\)	\(=\)	29.2
Dual form			29.3.f.a.15.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.58169 + 0.290886i) q^{2} +(-2.11401 - 1.32832i) q^{3} +(2.68078 + 0.611870i) q^{4} +(-2.49104 + 1.98654i) q^{5} +(-5.07133 - 4.04425i) q^{6} +(1.30161 + 5.70272i) q^{7} +(-3.06598 - 1.07283i) q^{8} +(-1.20034 - 2.49253i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 16 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} + 28 q^{8} - 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.58169	+	0.290886i	1.29084	+	0.145443i	0.730554	−	0.682854i	\(-0.239262\pi\)
							0.560289	+	0.828298i	\(0.310690\pi\)
\(3\)	−2.11401	−	1.32832i	−0.704671	−	0.442774i	0.131447	−	0.991323i	\(-0.458038\pi\)
							−0.836119	+	0.548549i	\(0.815180\pi\)
\(5\)	−2.49104	+	1.98654i	−0.498207	+	0.397307i	−0.840100	−	0.542432i	\(-0.817504\pi\)
							0.341892	+	0.939739i	\(0.388932\pi\)
\(7\)	1.30161	+	5.70272i	0.185944	+	0.814675i	0.978726	+	0.205170i	\(0.0657746\pi\)
							−0.792782	+	0.609505i	\(0.791368\pi\)
\(11\)	16.1070	−	5.63608i	1.46427	−	0.512371i	0.523682	−	0.851914i	\(-0.324558\pi\)
							0.940591	+	0.339543i	\(0.110272\pi\)
\(13\)	2.84196	−	5.90140i	0.218613	−	0.453954i	−0.762603	−	0.646866i	\(-0.776079\pi\)
							0.981216	+	0.192913i	\(0.0617934\pi\)
\(17\)	−14.7807	+	14.7807i	−0.869452	+	0.869452i	−0.992412	−	0.122959i	\(-0.960762\pi\)
							0.122959	+	0.992412i	\(0.460762\pi\)
\(19\)	9.65814	+	15.3708i	0.508323	+	0.808992i	0.998020	−	0.0629012i	\(-0.0200353\pi\)
							−0.489697	+	0.871893i	\(0.662892\pi\)
\(23\)	18.5750	−	23.2923i	0.807608	−	1.01271i	−0.191902	−	0.981414i	\(-0.561466\pi\)
							0.999510	−	0.0312943i	\(-0.00996291\pi\)
\(29\)	−19.9272	−	21.0691i	−0.687144	−	0.726521i
							\(31\)	−13.3687	−	1.50629i	−0.431249	−	0.0485900i	−0.106327	−	0.994331i	\(-0.533909\pi\)
−0.324921	+	0.945741i	\(0.605338\pi\)
\(37\)	17.2522	+	6.03682i	0.466277	+	0.163157i	0.553184	−	0.833059i	\(-0.313413\pi\)
							−0.0869071	+	0.996216i	\(0.527698\pi\)
\(41\)	−44.4102	−	44.4102i	−1.08318	−	1.08318i	−0.996211	−	0.0869640i	\(-0.972283\pi\)
							−0.0869640	−	0.996211i	\(-0.527717\pi\)
\(43\)	7.00239	+	62.1479i	0.162846	+	1.44530i	0.763509	+	0.645797i	\(0.223475\pi\)
							−0.600663	+	0.799503i	\(0.705096\pi\)
\(47\)	−16.6790	−	47.6658i	−0.354872	−	1.01417i	−0.973541	−	0.228515i	\(-0.926613\pi\)
							0.618668	−	0.785652i	\(-0.287673\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.3.f.a.2.4	✓	48
3.2	odd	2		261.3.s.a.118.1		48
29.15	odd	28	inner	29.3.f.a.15.4	yes	48
87.44	even	28		261.3.s.a.73.1		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.3.f.a.2.4	✓	48	1.1	even	1	trivial
29.3.f.a.15.4	yes	48	29.15	odd	28	inner
261.3.s.a.73.1		48	87.44	even	28
261.3.s.a.118.1		48	3.2	odd	2