Embedded newform 29.3.f.a.14.4

• Label: 29.3.f.a.14.4
• Level: $29$
• Weight: $3$
• Character: 29.14
• 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			14.4
Character	\(\chi\)	\(=\)	29.14
Dual form			29.3.f.a.27.4

$q$-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 \(\left(\mathbb{Z}/29\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{13}{28}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.3.f.a.14.4	✓	48	1.1	even	1	trivial
29.3.f.a.27.4	yes	48	29.27	odd	28	inner
261.3.s.a.172.1		48	87.56	even	28
261.3.s.a.217.1		48	3.2	odd	2