Embedded newform 29.3.f.a.3.3

• Label: 29.3.f.a.3.3
• Level: $29$
• Weight: $3$
• Character: 29.3
• 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			3.3
Character	\(\chi\)	\(=\)	29.3
Dual form			29.3.f.a.10.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.44470 + 0.907767i) q^{2} +(1.22940 - 0.430186i) q^{3} +(-0.472410 - 0.980970i) q^{4} +(-8.15497 + 1.86132i) q^{5} +(2.16663 + 0.494518i) q^{6} +(7.53715 + 3.62970i) q^{7} +(0.972146 - 8.62804i) q^{8} +(-5.71012 + 4.55367i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 16 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} + 28 q^{8} - 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.44470	+	0.907767i	0.722351	+	0.453883i	0.842378	−	0.538887i	\(-0.181155\pi\)
							−0.120027	+	0.992771i	\(0.538298\pi\)
\(3\)	1.22940	−	0.430186i	0.409800	−	0.143395i	−0.117503	−	0.993073i	\(-0.537489\pi\)
							0.527303	+	0.849677i	\(0.323203\pi\)
\(5\)	−8.15497	+	1.86132i	−1.63099	+	0.372264i	−0.937442	−	0.348142i	\(-0.886813\pi\)
							−0.693552	+	0.720406i	\(0.743955\pi\)
\(7\)	7.53715	+	3.62970i	1.07674	+	0.518528i	0.886272	−	0.463166i	\(-0.153287\pi\)
							0.190464	+	0.981694i	\(0.439001\pi\)
\(11\)	−1.08366	−	9.61773i	−0.0985143	−	0.874339i	−0.941640	−	0.336622i	\(-0.890716\pi\)
							0.843126	−	0.537717i	\(-0.180713\pi\)
\(13\)	8.88076	+	7.08217i	0.683135	+	0.544782i	0.902409	−	0.430881i	\(-0.141797\pi\)
							−0.219274	+	0.975663i	\(0.570369\pi\)
\(17\)	11.2928	−	11.2928i	0.664280	−	0.664280i	−0.292106	−	0.956386i	\(-0.594356\pi\)
							0.956386	+	0.292106i	\(0.0943559\pi\)
\(19\)	−5.31570	+	15.1914i	−0.279774	+	0.799548i	0.715155	+	0.698966i	\(0.246356\pi\)
							−0.994929	+	0.100582i	\(0.967930\pi\)
\(23\)	−2.98784	+	13.0906i	−0.129906	+	0.569155i	0.867517	+	0.497408i	\(0.165715\pi\)
							−0.997423	+	0.0717472i	\(0.977143\pi\)
\(29\)	−25.9185	−	13.0088i	−0.893743	−	0.448580i
							\(31\)	1.32569	+	0.832985i	0.0427641	+	0.0268705i	0.553244	−	0.833019i	\(-0.313390\pi\)
−0.510480	+	0.859889i	\(0.670532\pi\)
\(37\)	2.36887	−	21.0243i	0.0640236	−	0.568225i	−0.920219	−	0.391403i	\(-0.871990\pi\)
							0.984243	−	0.176822i	\(-0.0565817\pi\)
\(41\)	−0.193896	−	0.193896i	−0.00472917	−	0.00472917i	0.704738	−	0.709467i	\(-0.251064\pi\)
							−0.709467	+	0.704738i	\(0.751064\pi\)
\(43\)	−38.8303	−	61.7981i	−0.903031	−	1.43717i	−0.899378	−	0.437172i	\(-0.855980\pi\)
							−0.00365282	−	0.999993i	\(-0.501163\pi\)
\(47\)	42.6067	−	4.80062i	0.906525	−	0.102141i	0.353624	−	0.935388i	\(-0.384949\pi\)
							0.552901	+	0.833247i	\(0.313521\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.3.f.a.3.3	✓	48
3.2	odd	2		261.3.s.a.235.2		48
29.10	odd	28	inner	29.3.f.a.10.3	yes	48
87.68	even	28		261.3.s.a.10.2		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.3.f.a.3.3	✓	48	1.1	even	1	trivial
29.3.f.a.10.3	yes	48	29.10	odd	28	inner
261.3.s.a.10.2		48	87.68	even	28
261.3.s.a.235.2		48	3.2	odd	2