Embedded newform 48.5.i.a.5.15

• Label: 48.5.i.a.5.15
• Level: $48$
• Weight: $5$
• Character: 48.5
• Analytic conductor: $4.962$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	48.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.96175822802\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(30\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			5.15
Character	\(\chi\)	\(=\)	48.5
Dual form			48.5.i.a.29.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.312407 - 3.98778i) q^{2} +(-8.85016 - 1.63546i) q^{3} +(-15.8048 + 2.49162i) q^{4} +(2.85930 - 2.85930i) q^{5} +(-3.75701 + 35.8034i) q^{6} +27.2580i q^{7} +(14.8736 + 62.2477i) q^{8} +(75.6505 + 28.9482i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 2 q^{3} - 4 q^{4} - 68 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\)	\(17\)	\(31\)	\(37\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.312407	−	3.98778i	−0.0781017	−	0.996945i
							\(3\)	−8.85016	−	1.63546i	−0.983351	−	0.181718i
\(5\)	2.85930	−	2.85930i	0.114372	−	0.114372i								−0.647604	−	0.761977i	\(-0.724229\pi\)
							0.761977	+	0.647604i	\(0.224229\pi\)
\(7\)			27.2580i			0.556286i	0.960540	+	0.278143i	\(0.0897189\pi\)
							−0.960540	+	0.278143i	\(0.910281\pi\)
\(11\)	−77.0090	+	77.0090i	−0.636438	+	0.636438i	−0.949675	−	0.313237i	\(-0.898587\pi\)
							0.313237	+	0.949675i	\(0.398587\pi\)
\(13\)	−63.4325	+	63.4325i	−0.375340	+	0.375340i	−0.869418	−	0.494077i	\(-0.835506\pi\)
							0.494077	+	0.869418i	\(0.335506\pi\)
\(17\)		−	161.958i		−	0.560409i	−0.959940	−	0.280204i	\(-0.909598\pi\)
							0.959940	−	0.280204i	\(-0.0904022\pi\)
\(19\)	−390.729	+	390.729i	−1.08235	+	1.08235i	−0.0860615	+	0.996290i	\(0.527428\pi\)
							−0.996290	+	0.0860615i	\(0.972572\pi\)
\(23\)	−720.957			−1.36287			−0.681434	−	0.731880i	\(-0.738643\pi\)
							−0.681434	+	0.731880i	\(0.738643\pi\)
\(29\)	−693.034	−	693.034i	−0.824060	−	0.824060i	0.162628	−	0.986687i	\(-0.448003\pi\)
							−0.986687	+	0.162628i	\(0.948003\pi\)
\(31\)	786.880			0.818814			0.409407	−	0.912352i	\(-0.365736\pi\)
							0.409407	+	0.912352i	\(0.365736\pi\)
\(37\)	538.459	+	538.459i	0.393323	+	0.393323i	0.875870	−	0.482547i	\(-0.160288\pi\)
							−0.482547	+	0.875870i	\(0.660288\pi\)
\(41\)	925.465			0.550544			0.275272	−	0.961366i	\(-0.411232\pi\)
							0.275272	+	0.961366i	\(0.411232\pi\)
\(43\)	−1599.27	−	1599.27i	−0.864935	−	0.864935i	0.126971	−	0.991906i	\(-0.459474\pi\)
							−0.991906	+	0.126971i	\(0.959474\pi\)
\(47\)			3186.64i			1.44257i	0.692637	+	0.721286i	\(0.256449\pi\)
							−0.692637	+	0.721286i	\(0.743551\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.5.i.a.5.15	✓	60
3.2	odd	2	inner	48.5.i.a.5.16	yes	60
4.3	odd	2		192.5.i.a.113.28		60
12.11	even	2		192.5.i.a.113.13		60
16.3	odd	4		192.5.i.a.17.13		60
16.13	even	4	inner	48.5.i.a.29.16	yes	60
48.29	odd	4	inner	48.5.i.a.29.15	yes	60
48.35	even	4		192.5.i.a.17.28		60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.5.i.a.5.15	✓	60	1.1	even	1	trivial
48.5.i.a.5.16	yes	60	3.2	odd	2	inner
48.5.i.a.29.15	yes	60	48.29	odd	4	inner
48.5.i.a.29.16	yes	60	16.13	even	4	inner
192.5.i.a.17.13		60	16.3	odd	4
192.5.i.a.17.28		60	48.35	even	4
192.5.i.a.113.13		60	12.11	even	2
192.5.i.a.113.28		60	4.3	odd	2