Embedded newform 48.9.l.a.19.32

• Label: 48.9.l.a.19.32
• Level: $48$
• Weight: $9$
• Character: 48.19
• 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			19.32
Character	\(\chi\)	\(=\)	48.19
Dual form			48.9.l.a.43.32

$q$-expansion

\(f(q)\)	\(=\)	\(q+(15.7988 + 2.52939i) q^{2} +(33.0681 - 33.0681i) q^{3} +(243.204 + 79.9227i) q^{4} +(-384.834 + 384.834i) q^{5} +(606.079 - 438.794i) q^{6} +1641.15 q^{7} +(3640.18 + 1877.84i) q^{8} -2187.00i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 372 q^{4} + 17460 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\)	\(17\)	\(31\)	\(37\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	15.7988	+	2.52939i	0.987425	+	0.158087i
							\(3\)	33.0681	−	33.0681i	0.408248	−	0.408248i
\(5\)	−384.834	+	384.834i	−0.615734	+	0.615734i								−0.944434	−	0.328700i	\(-0.893389\pi\)
							0.328700	+	0.944434i	\(0.393389\pi\)
\(7\)	1641.15			0.683528			0.341764	−	0.939786i	\(-0.388976\pi\)
							0.341764	+	0.939786i	\(0.388976\pi\)
\(11\)	5395.29	+	5395.29i	0.368506	+	0.368506i	0.866932	−	0.498426i	\(-0.166089\pi\)
							−0.498426	+	0.866932i	\(0.666089\pi\)
\(13\)	31216.1	+	31216.1i	1.09296	+	1.09296i	0.995211	+	0.0977498i	\(0.0311645\pi\)
							0.0977498	+	0.995211i	\(0.468836\pi\)
\(17\)	−103035.			−1.23364			−0.616821	−	0.787103i	\(-0.711580\pi\)
							−0.616821	+	0.787103i	\(0.711580\pi\)
\(19\)	143933.	−	143933.i	1.10445	−	1.10445i	0.110584	−	0.993867i	\(-0.464728\pi\)
							0.993867	−	0.110584i	\(-0.0352722\pi\)
\(23\)	322047.			1.15082			0.575410	−	0.817865i	\(-0.304842\pi\)
							0.575410	+	0.817865i	\(0.304842\pi\)
\(29\)	−645577.	−	645577.i	−0.912759	−	0.912759i	0.0837291	−	0.996489i	\(-0.473317\pi\)
							−0.996489	+	0.0837291i	\(0.973317\pi\)
\(31\)			611529.i			0.662171i	0.943601	+	0.331086i	\(0.107415\pi\)
							−0.943601	+	0.331086i	\(0.892585\pi\)
\(37\)	−486387.	+	486387.i	−0.259523	+	0.259523i	−0.824860	−	0.565337i	\(-0.808746\pi\)
							0.565337	+	0.824860i	\(0.308746\pi\)
\(41\)			852590.i			0.301720i	0.988555	+	0.150860i	\(0.0482043\pi\)
							−0.988555	+	0.150860i	\(0.951796\pi\)
\(43\)	−821316.	−	821316.i	−0.240235	−	0.240235i	0.576712	−	0.816947i	\(-0.304335\pi\)
							−0.816947	+	0.576712i	\(0.804335\pi\)
\(47\)		−	2.87783e6i		−	0.589758i	−0.955535	−	0.294879i	\(-0.904721\pi\)
							0.955535	−	0.294879i	\(-0.0952794\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.9.l.a.19.32	✓	64
4.3	odd	2		192.9.l.a.175.5		64
16.5	even	4		192.9.l.a.79.5		64
16.11	odd	4	inner	48.9.l.a.43.32	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.9.l.a.19.32	✓	64	1.1	even	1	trivial
48.9.l.a.43.32	yes	64	16.11	odd	4	inner
192.9.l.a.79.5		64	16.5	even	4
192.9.l.a.175.5		64	4.3	odd	2