Embedded newform 48.5.i.a.5.19

• Label: 48.5.i.a.5.19
• 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			5.19
Character	\(\chi\)	\(=\)	48.5
Dual form			48.5.i.a.29.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.96807 - 3.48234i) q^{2} +(0.667113 - 8.97524i) q^{3} +(-8.25342 - 13.7070i) q^{4} +(17.7419 - 17.7419i) q^{5} +(-29.9419 - 19.9870i) q^{6} +66.4694i q^{7} +(-63.9757 + 1.76496i) q^{8} +(-80.1099 - 11.9750i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 2 q^{3} - 4 q^{4} - 68 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.96807	−	3.48234i	0.492017	−	0.870586i
							\(3\)	0.667113	−	8.97524i	0.0741236	−	0.997249i
\(5\)	17.7419	−	17.7419i	0.709677	−	0.709677i								−0.256790	−	0.966467i	\(-0.582665\pi\)
							0.966467	+	0.256790i	\(0.0826649\pi\)
\(7\)			66.4694i			1.35652i	0.734823	+	0.678259i	\(0.237265\pi\)
							−0.734823	+	0.678259i	\(0.762735\pi\)
\(11\)	124.446	−	124.446i	1.02848	−	1.02848i	0.0288939	−	0.999582i	\(-0.490802\pi\)
							0.999582	−	0.0288939i	\(-0.00919849\pi\)
\(13\)	−64.0011	+	64.0011i	−0.378705	+	0.378705i	−0.870635	−	0.491930i	\(-0.836292\pi\)
							0.491930	+	0.870635i	\(0.336292\pi\)
\(17\)			74.1965i			0.256735i	0.991727	+	0.128368i	\(0.0409738\pi\)
							−0.991727	+	0.128368i	\(0.959026\pi\)
\(19\)	442.958	−	442.958i	1.22703	−	1.22703i	0.261947	−	0.965082i	\(-0.415635\pi\)
							0.965082	−	0.261947i	\(-0.0843646\pi\)
\(23\)	181.796			0.343660			0.171830	−	0.985127i	\(-0.445032\pi\)
							0.171830	+	0.985127i	\(0.445032\pi\)
\(29\)	−320.985	−	320.985i	−0.381671	−	0.381671i	0.490033	−	0.871704i	\(-0.336985\pi\)
							−0.871704	+	0.490033i	\(0.836985\pi\)
\(31\)	1493.03			1.55362			0.776811	−	0.629734i	\(-0.216836\pi\)
							0.776811	+	0.629734i	\(0.216836\pi\)
\(37\)	490.065	+	490.065i	0.357973	+	0.357973i	0.863065	−	0.505092i	\(-0.168542\pi\)
							−0.505092	+	0.863065i	\(0.668542\pi\)
\(41\)	−1104.51			−0.657054			−0.328527	−	0.944495i	\(-0.606552\pi\)
							−0.328527	+	0.944495i	\(0.606552\pi\)
\(43\)	−1274.62	−	1274.62i	−0.689359	−	0.689359i	0.272732	−	0.962090i	\(-0.412073\pi\)
							−0.962090	+	0.272732i	\(0.912073\pi\)
\(47\)			2133.60i			0.965868i	0.875657	+	0.482934i	\(0.160429\pi\)
							−0.875657	+	0.482934i	\(0.839571\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.5.i.a.5.19	yes	60
3.2	odd	2	inner	48.5.i.a.5.12	✓	60
4.3	odd	2		192.5.i.a.113.15		60
12.11	even	2		192.5.i.a.113.1		60
16.3	odd	4		192.5.i.a.17.1		60
16.13	even	4	inner	48.5.i.a.29.12	yes	60
48.29	odd	4	inner	48.5.i.a.29.19	yes	60
48.35	even	4		192.5.i.a.17.15		60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.5.i.a.5.12	✓	60	3.2	odd	2	inner
48.5.i.a.5.19	yes	60	1.1	even	1	trivial
48.5.i.a.29.12	yes	60	16.13	even	4	inner
48.5.i.a.29.19	yes	60	48.29	odd	4	inner
192.5.i.a.17.1		60	16.3	odd	4
192.5.i.a.17.15		60	48.35	even	4
192.5.i.a.113.1		60	12.11	even	2
192.5.i.a.113.15		60	4.3	odd	2