Embedded newform 48.9.l.a.43.29

• Label: 48.9.l.a.43.29
• Level: $48$
• Weight: $9$
• Character: 48.43
• Analytic conductor: $19.554$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	48.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.5541732829\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			43.29
Character	\(\chi\)	\(=\)	48.43
Dual form			48.9.l.a.19.29

$q$-expansion

\(f(q)\)	\(=\)	\(q+(14.8971 - 5.83746i) q^{2} +(33.0681 + 33.0681i) q^{3} +(187.848 - 173.923i) q^{4} +(625.993 + 625.993i) q^{5} +(685.653 + 299.586i) q^{6} +873.478 q^{7} +(1783.13 - 3687.50i) q^{8} +2187.00i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 372 q^{4} + 17460 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	14.8971	−	5.83746i	0.931070	−	0.364841i
							\(3\)	33.0681	+	33.0681i	0.408248	+	0.408248i
\(5\)	625.993	+	625.993i	1.00159	+	1.00159i								0.999999	+	0.00159026i	\(0.000506195\pi\)
							0.00159026	+	0.999999i	\(0.499494\pi\)
\(7\)	873.478			0.363798			0.181899	−	0.983317i	\(-0.441776\pi\)
							0.181899	+	0.983317i	\(0.441776\pi\)
\(11\)	−7493.81	+	7493.81i	−0.511837	+	0.511837i	−0.915089	−	0.403252i	\(-0.867880\pi\)
							0.403252	+	0.915089i	\(0.367880\pi\)
\(13\)	−17607.0	+	17607.0i	−0.616469	+	0.616469i	−0.944624	−	0.328155i	\(-0.893573\pi\)
							0.328155	+	0.944624i	\(0.393573\pi\)
\(17\)	120211.			1.43929			0.719645	−	0.694342i	\(-0.244304\pi\)
							0.719645	+	0.694342i	\(0.244304\pi\)
\(19\)	−20666.5	−	20666.5i	−0.158582	−	0.158582i	0.623356	−	0.781938i	\(-0.285769\pi\)
							−0.781938	+	0.623356i	\(0.785769\pi\)
\(23\)	−4997.49			−0.0178583			−0.00892916	−	0.999960i	\(-0.502842\pi\)
							−0.00892916	+	0.999960i	\(0.502842\pi\)
\(29\)	591450.	−	591450.i	0.836231	−	0.836231i	−0.152130	−	0.988361i	\(-0.548613\pi\)
							0.988361	+	0.152130i	\(0.0486132\pi\)
\(31\)		−	499319.i		−	0.540669i	−0.962766	−	0.270334i	\(-0.912866\pi\)
							0.962766	−	0.270334i	\(-0.0871343\pi\)
\(37\)	−548610.	−	548610.i	−0.292723	−	0.292723i	0.545432	−	0.838155i	\(-0.316366\pi\)
							−0.838155	+	0.545432i	\(0.816366\pi\)
\(41\)			2.47126e6i			0.874546i	0.899329	+	0.437273i	\(0.144056\pi\)
							−0.899329	+	0.437273i	\(0.855944\pi\)
\(43\)	1.86856e6	−	1.86856e6i	0.546555	−	0.546555i	−0.378888	−	0.925443i	\(-0.623693\pi\)
							0.925443	+	0.378888i	\(0.123693\pi\)
\(47\)			3.07567e6i			0.630301i	0.949042	+	0.315150i	\(0.102055\pi\)
							−0.949042	+	0.315150i	\(0.897945\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.9.l.a.43.29	yes	64
4.3	odd	2		192.9.l.a.79.15		64
16.3	odd	4	inner	48.9.l.a.19.29	✓	64
16.13	even	4		192.9.l.a.175.15		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.9.l.a.19.29	✓	64	16.3	odd	4	inner
48.9.l.a.43.29	yes	64	1.1	even	1	trivial
192.9.l.a.79.15		64	4.3	odd	2
192.9.l.a.175.15		64	16.13	even	4