Embedded newform 48.5.i.a.5.9

• Label: 48.5.i.a.5.9
• 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			5.9
Character	\(\chi\)	\(=\)	48.5
Dual form			48.5.i.a.29.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.23278 - 3.31884i) q^{2} +(8.02371 + 4.07678i) q^{3} +(-6.02942 + 14.8205i) q^{4} +(12.9156 - 12.9156i) q^{5} +(-4.38496 - 35.7319i) q^{6} +84.6751i q^{7} +(62.6491 - 13.0801i) q^{8} +(47.7598 + 65.4217i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 2 q^{3} - 4 q^{4} - 68 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.23278	−	3.31884i	−0.558194	−	0.829711i
							\(3\)	8.02371	+	4.07678i	0.891523	+	0.452975i
\(5\)	12.9156	−	12.9156i	0.516623	−	0.516623i								−0.399925	−	0.916548i	\(-0.630964\pi\)
							0.916548	+	0.399925i	\(0.130964\pi\)
\(7\)			84.6751i			1.72806i	0.503438	+	0.864032i	\(0.332068\pi\)
							−0.503438	+	0.864032i	\(0.667932\pi\)
\(11\)	43.1897	−	43.1897i	0.356940	−	0.356940i	−0.505744	−	0.862684i	\(-0.668782\pi\)
							0.862684	+	0.505744i	\(0.168782\pi\)
\(13\)	202.110	−	202.110i	1.19592	−	1.19592i	0.220539	−	0.975378i	\(-0.429218\pi\)
							0.975378	−	0.220539i	\(-0.0707818\pi\)
\(17\)			114.001i			0.394467i	0.980357	+	0.197233i	\(0.0631957\pi\)
							−0.980357	+	0.197233i	\(0.936804\pi\)
\(19\)	−144.349	+	144.349i	−0.399860	+	0.399860i	−0.878184	−	0.478324i	\(-0.841245\pi\)
							0.478324	+	0.878184i	\(0.341245\pi\)
\(23\)	150.265			0.284054			0.142027	−	0.989863i	\(-0.454638\pi\)
							0.142027	+	0.989863i	\(0.454638\pi\)
\(29\)	−992.052	−	992.052i	−1.17961	−	1.17961i	−0.979844	−	0.199766i	\(-0.935982\pi\)
							−0.199766	−	0.979844i	\(-0.564018\pi\)
\(31\)	41.6872			0.0433790			0.0216895	−	0.999765i	\(-0.493095\pi\)
							0.0216895	+	0.999765i	\(0.493095\pi\)
\(37\)	−679.805	−	679.805i	−0.496571	−	0.496571i	0.413798	−	0.910369i	\(-0.364202\pi\)
							−0.910369	+	0.413798i	\(0.864202\pi\)
\(41\)	−1814.04			−1.07914			−0.539572	−	0.841939i	\(-0.681414\pi\)
							−0.539572	+	0.841939i	\(0.681414\pi\)
\(43\)	662.537	+	662.537i	0.358322	+	0.358322i	0.863194	−	0.504872i	\(-0.168460\pi\)
							−0.504872	+	0.863194i	\(0.668460\pi\)
\(47\)		−	2860.57i		−	1.29496i	−0.762082	−	0.647480i	\(-0.775823\pi\)
							0.762082	−	0.647480i	\(-0.224177\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.5.i.a.5.9	✓	60
3.2	odd	2	inner	48.5.i.a.5.22	yes	60
4.3	odd	2		192.5.i.a.113.5		60
12.11	even	2		192.5.i.a.113.19		60
16.3	odd	4		192.5.i.a.17.19		60
16.13	even	4	inner	48.5.i.a.29.22	yes	60
48.29	odd	4	inner	48.5.i.a.29.9	yes	60
48.35	even	4		192.5.i.a.17.5		60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.5.i.a.5.9	✓	60	1.1	even	1	trivial
48.5.i.a.5.22	yes	60	3.2	odd	2	inner
48.5.i.a.29.9	yes	60	48.29	odd	4	inner
48.5.i.a.29.22	yes	60	16.13	even	4	inner
192.5.i.a.17.5		60	48.35	even	4
192.5.i.a.17.19		60	16.3	odd	4
192.5.i.a.113.5		60	4.3	odd	2
192.5.i.a.113.19		60	12.11	even	2