Embedded newform 48.5.i.a.5.16

• Label: 48.5.i.a.5.16
• 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			5.16
Character	\(\chi\)	\(=\)	48.5
Dual form			48.5.i.a.29.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.312407 + 3.98778i) q^{2} +(1.63546 + 8.85016i) q^{3} +(-15.8048 + 2.49162i) q^{4} +(-2.85930 + 2.85930i) q^{5} +(-34.7816 + 9.28671i) q^{6} +27.2580i q^{7} +(-14.8736 - 62.2477i) q^{8} +(-75.6505 + 28.9482i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 2 q^{3} - 4 q^{4} - 68 q^{6}+O(q^{10}) \)

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
							\(3\)	1.63546	+	8.85016i	0.181718	+	0.983351i
\(5\)	−2.85930	+	2.85930i	−0.114372	+	0.114372i								−0.761977	−	0.647604i	\(-0.775771\pi\)
							0.647604	+	0.761977i	\(0.275771\pi\)
\(7\)			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	−0.313237	−	0.949675i	\(-0.601413\pi\)
							0.949675	+	0.313237i	\(0.101413\pi\)
\(13\)	−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	\(0.0904022\pi\)
							−0.959940	+	0.280204i	\(0.909598\pi\)
\(19\)	−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	\(-0.261357\pi\)
							0.681434	+	0.731880i	\(0.261357\pi\)
\(29\)	693.034	+	693.034i	0.824060	+	0.824060i	0.986687	−	0.162628i	\(-0.0519970\pi\)
							−0.162628	+	0.986687i	\(0.551997\pi\)
\(31\)	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	\(-0.588768\pi\)
							−0.275272	+	0.961366i	\(0.588768\pi\)
\(43\)	−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	\(-0.743551\pi\)
							0.692637	−	0.721286i	\(-0.256449\pi\)

(See \(a_n\) instead)

Twists

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

    

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