Embedded newform 48.5.i.a.5.8

• Label: 48.5.i.a.5.8
• 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			5.8
Character	\(\chi\)	\(=\)	48.5
Dual form			48.5.i.a.29.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.98404 + 2.66374i) q^{2} +(6.71421 + 5.99328i) q^{3} +(1.80902 - 15.8974i) q^{4} +(27.3100 - 27.3100i) q^{5} +(-36.0000 + 0.000682234i) q^{6} -70.5035i q^{7} +(36.9483 + 52.2573i) q^{8} +(9.16120 + 80.4803i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 2 q^{3} - 4 q^{4} - 68 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.98404	+	2.66374i	−0.746011	+	0.665934i
							\(3\)	6.71421	+	5.99328i	0.746023	+	0.665920i
\(5\)	27.3100	−	27.3100i	1.09240	−	1.09240i								0.0971278	−	0.995272i	\(-0.469034\pi\)
							0.995272	−	0.0971278i	\(-0.0309656\pi\)
\(7\)		−	70.5035i		−	1.43885i	−0.694571	−	0.719424i	\(-0.744406\pi\)
							0.694571	−	0.719424i	\(-0.255594\pi\)
\(11\)	30.6516	−	30.6516i	0.253319	−	0.253319i	−0.569011	−	0.822330i	\(-0.692674\pi\)
							0.822330	+	0.569011i	\(0.192674\pi\)
\(13\)	48.1898	−	48.1898i	0.285147	−	0.285147i	−0.550011	−	0.835158i	\(-0.685376\pi\)
							0.835158	+	0.550011i	\(0.185376\pi\)
\(17\)			508.352i			1.75900i	0.475895	+	0.879502i	\(0.342124\pi\)
							−0.475895	+	0.879502i	\(0.657876\pi\)
\(19\)	145.442	−	145.442i	0.402885	−	0.402885i	−0.476363	−	0.879249i	\(-0.658045\pi\)
							0.879249	+	0.476363i	\(0.158045\pi\)
\(23\)	−395.865			−0.748327			−0.374163	−	0.927363i	\(-0.622070\pi\)
							−0.374163	+	0.927363i	\(0.622070\pi\)
\(29\)	−135.024	−	135.024i	−0.160551	−	0.160551i	0.622260	−	0.782811i	\(-0.286215\pi\)
							−0.782811	+	0.622260i	\(0.786215\pi\)
\(31\)	−266.252			−0.277058			−0.138529	−	0.990358i	\(-0.544237\pi\)
							−0.138529	+	0.990358i	\(0.544237\pi\)
\(37\)	1484.91	+	1484.91i	1.08467	+	1.08467i	0.996067	+	0.0885993i	\(0.0282391\pi\)
							0.0885993	+	0.996067i	\(0.471761\pi\)
\(41\)	1756.02			1.04463			0.522313	−	0.852754i	\(-0.325069\pi\)
							0.522313	+	0.852754i	\(0.325069\pi\)
\(43\)	−908.971	−	908.971i	−0.491601	−	0.491601i	0.417209	−	0.908810i	\(-0.363008\pi\)
							−0.908810	+	0.417209i	\(0.863008\pi\)
\(47\)			155.961i			0.0706027i	0.999377	+	0.0353013i	\(0.0112391\pi\)
							−0.999377	+	0.0353013i	\(0.988761\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.5.i.a.5.8	✓	60
3.2	odd	2	inner	48.5.i.a.5.23	yes	60
4.3	odd	2		192.5.i.a.113.8		60
12.11	even	2		192.5.i.a.113.23		60
16.3	odd	4		192.5.i.a.17.23		60
16.13	even	4	inner	48.5.i.a.29.23	yes	60
48.29	odd	4	inner	48.5.i.a.29.8	yes	60
48.35	even	4		192.5.i.a.17.8		60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.5.i.a.5.8	✓	60	1.1	even	1	trivial
48.5.i.a.5.23	yes	60	3.2	odd	2	inner
48.5.i.a.29.8	yes	60	48.29	odd	4	inner
48.5.i.a.29.23	yes	60	16.13	even	4	inner
192.5.i.a.17.8		60	48.35	even	4
192.5.i.a.17.23		60	16.3	odd	4
192.5.i.a.113.8		60	4.3	odd	2
192.5.i.a.113.23		60	12.11	even	2