Embedded newform 29.3.f.a.27.4

• Label: 29.3.f.a.27.4
• Level: $29$
• Weight: $3$
• Character: 29.27
• 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			27.4
Character	\(\chi\)	\(=\)	29.27
Dual form			29.3.f.a.14.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.293128 - 2.60158i) q^{2} +(0.0782835 - 0.124587i) q^{3} +(-2.78259 - 0.635107i) q^{4} +(-1.65260 + 1.31790i) q^{5} +(-0.301177 - 0.240181i) q^{6} +(0.747987 + 3.27715i) q^{7} +(0.990802 - 2.83155i) q^{8} +(3.89556 + 8.08921i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 16 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} + 28 q^{8} - 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{15}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.293128	−	2.60158i	0.146564	−	1.30079i	−0.678017	−	0.735046i	\(-0.737160\pi\)
							0.824581	−	0.565744i	\(-0.191411\pi\)
\(3\)	0.0782835	−	0.124587i	0.0260945	−	0.0415291i	−0.833422	−	0.552637i	\(-0.813622\pi\)
							0.859517	+	0.511107i	\(0.170765\pi\)
\(5\)	−1.65260	+	1.31790i	−0.330520	+	0.263581i	−0.774662	−	0.632375i	\(-0.782080\pi\)
							0.444142	+	0.895956i	\(0.353509\pi\)
\(7\)	0.747987	+	3.27715i	0.106855	+	0.468164i	0.999837	+	0.0180702i	\(0.00575225\pi\)
							−0.892981	+	0.450094i	\(0.851391\pi\)
\(11\)	−0.536371	−	1.53286i	−0.0487610	−	0.139351i	0.916928	−	0.399053i	\(-0.130661\pi\)
							−0.965689	+	0.259703i	\(0.916375\pi\)
\(13\)	−3.68793	+	7.65807i	−0.283687	+	0.589082i	−0.993307	−	0.115502i	\(-0.963152\pi\)
							0.709620	+	0.704584i	\(0.248867\pi\)
\(17\)	−19.7801	−	19.7801i	−1.16353	−	1.16353i	−0.983696	−	0.179837i	\(-0.942443\pi\)
							−0.179837	−	0.983696i	\(-0.557557\pi\)
\(19\)	−9.28934	+	5.83688i	−0.488913	+	0.307204i	−0.753833	−	0.657066i	\(-0.771797\pi\)
							0.264920	+	0.964270i	\(0.414654\pi\)
\(23\)	20.8669	−	26.1663i	0.907259	−	1.13767i	−0.0827374	−	0.996571i	\(-0.526366\pi\)
							0.989996	−	0.141095i	\(-0.0450623\pi\)
\(29\)	12.6644	+	26.0886i	0.436703	+	0.899606i
							\(31\)	3.96474	−	35.1880i	0.127895	−	1.13510i	−0.751744	−	0.659455i	\(-0.770787\pi\)
0.879639	−	0.475642i	\(-0.157784\pi\)
\(37\)	2.28364	−	6.52628i	0.0617201	−	0.176386i	−0.908894	−	0.417027i	\(-0.863072\pi\)
							0.970614	+	0.240641i	\(0.0773577\pi\)
\(41\)	−22.9886	+	22.9886i	−0.560698	+	0.560698i	−0.929506	−	0.368808i	\(-0.879766\pi\)
							0.368808	+	0.929506i	\(0.379766\pi\)
\(43\)	55.6405	−	6.26918i	1.29397	−	0.145795i	0.562000	−	0.827137i	\(-0.310032\pi\)
							0.731965	+	0.681342i	\(0.238603\pi\)
\(47\)	60.5913	−	21.2018i	1.28918	−	0.451103i	0.403319	−	0.915059i	\(-0.367856\pi\)
							0.885858	+	0.463957i	\(0.153571\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.3.f.a.27.4	yes	48
3.2	odd	2		261.3.s.a.172.1		48
29.14	odd	28	inner	29.3.f.a.14.4	✓	48
87.14	even	28		261.3.s.a.217.1		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.3.f.a.14.4	✓	48	29.14	odd	28	inner
29.3.f.a.27.4	yes	48	1.1	even	1	trivial
261.3.s.a.172.1		48	3.2	odd	2
261.3.s.a.217.1		48	87.14	even	28