Embedded newform 48.5.i.a.5.13

• Label: 48.5.i.a.5.13
• 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			5.13
Character	\(\chi\)	\(=\)	48.5
Dual form			48.5.i.a.29.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.13167 + 3.83658i) q^{2} +(-8.98791 - 0.466380i) q^{3} +(-13.4387 - 8.68346i) q^{4} +(23.5639 - 23.5639i) q^{5} +(11.9606 - 33.9550i) q^{6} +30.4899i q^{7} +(48.5229 - 41.7317i) q^{8} +(80.5650 + 8.38356i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 2 q^{3} - 4 q^{4} - 68 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.13167	+	3.83658i	−0.282917	+	0.959144i
							\(3\)	−8.98791	−	0.466380i	−0.998656	−	0.0518200i
\(5\)	23.5639	−	23.5639i	0.942556	−	0.942556i								−0.0558818	−	0.998437i	\(-0.517797\pi\)
							0.998437	+	0.0558818i	\(0.0177970\pi\)
\(7\)			30.4899i			0.622242i	0.950370	+	0.311121i	\(0.100704\pi\)
							−0.950370	+	0.311121i	\(0.899296\pi\)
\(11\)	22.5385	−	22.5385i	0.186269	−	0.186269i	−0.607812	−	0.794081i	\(-0.707953\pi\)
							0.794081	+	0.607812i	\(0.207953\pi\)
\(13\)	86.0911	−	86.0911i	0.509415	−	0.509415i	−0.404932	−	0.914347i	\(-0.632705\pi\)
							0.914347	+	0.404932i	\(0.132705\pi\)
\(17\)		−	406.437i		−	1.40635i	−0.711014	−	0.703177i	\(-0.751764\pi\)
							0.711014	−	0.703177i	\(-0.248236\pi\)
\(19\)	340.364	−	340.364i	0.942836	−	0.942836i	−0.0556164	−	0.998452i	\(-0.517712\pi\)
							0.998452	+	0.0556164i	\(0.0177124\pi\)
\(23\)	549.957			1.03962			0.519808	−	0.854283i	\(-0.326003\pi\)
							0.519808	+	0.854283i	\(0.326003\pi\)
\(29\)	−190.284	−	190.284i	−0.226259	−	0.226259i	0.584869	−	0.811128i	\(-0.301146\pi\)
							−0.811128	+	0.584869i	\(0.801146\pi\)
\(31\)	−1712.72			−1.78223			−0.891113	−	0.453781i	\(-0.850075\pi\)
							−0.891113	+	0.453781i	\(0.850075\pi\)
\(37\)	1227.54	+	1227.54i	0.896669	+	0.896669i	0.995140	−	0.0984713i	\(-0.0313953\pi\)
							−0.0984713	+	0.995140i	\(0.531395\pi\)
\(41\)	−1560.26			−0.928175			−0.464087	−	0.885789i	\(-0.653618\pi\)
							−0.464087	+	0.885789i	\(0.653618\pi\)
\(43\)	1290.04	+	1290.04i	0.697695	+	0.697695i	0.963913	−	0.266218i	\(-0.0857741\pi\)
							−0.266218	+	0.963913i	\(0.585774\pi\)
\(47\)		−	1736.02i		−	0.785884i	−0.919563	−	0.392942i	\(-0.871457\pi\)
							0.919563	−	0.392942i	\(-0.128543\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.5.i.a.5.13	✓	60
3.2	odd	2	inner	48.5.i.a.5.18	yes	60
4.3	odd	2		192.5.i.a.113.30		60
12.11	even	2		192.5.i.a.113.16		60
16.3	odd	4		192.5.i.a.17.16		60
16.13	even	4	inner	48.5.i.a.29.18	yes	60
48.29	odd	4	inner	48.5.i.a.29.13	yes	60
48.35	even	4		192.5.i.a.17.30		60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.5.i.a.5.13	✓	60	1.1	even	1	trivial
48.5.i.a.5.18	yes	60	3.2	odd	2	inner
48.5.i.a.29.13	yes	60	48.29	odd	4	inner
48.5.i.a.29.18	yes	60	16.13	even	4	inner
192.5.i.a.17.16		60	16.3	odd	4
192.5.i.a.17.30		60	48.35	even	4
192.5.i.a.113.16		60	12.11	even	2
192.5.i.a.113.30		60	4.3	odd	2