Embedded newform 48.9.l.a.43.2

• Label: 48.9.l.a.43.2
• 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			43.2
Character	\(\chi\)	\(=\)	48.43
Dual form			48.9.l.a.19.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-15.6102 - 3.51009i) q^{2} +(-33.0681 - 33.0681i) q^{3} +(231.359 + 109.587i) q^{4} +(801.254 + 801.254i) q^{5} +(400.129 + 632.273i) q^{6} +3981.15 q^{7} +(-3226.90 - 2522.76i) q^{8} +2187.00i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 372 q^{4} + 17460 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−15.6102	−	3.51009i	−0.975639	−	0.219381i
							\(3\)	−33.0681	−	33.0681i	−0.408248	−	0.408248i
\(5\)	801.254	+	801.254i	1.28201	+	1.28201i								0.939524	+	0.342483i	\(0.111268\pi\)
							0.342483	+	0.939524i	\(0.388732\pi\)
\(7\)	3981.15			1.65812			0.829061	−	0.559158i	\(-0.188876\pi\)
							0.829061	+	0.559158i	\(0.188876\pi\)
\(11\)	−11803.7	+	11803.7i	−0.806206	+	0.806206i	−0.984057	−	0.177851i	\(-0.943085\pi\)
							0.177851	+	0.984057i	\(0.443085\pi\)
\(13\)	−9800.60	+	9800.60i	−0.343146	+	0.343146i	−0.857549	−	0.514402i	\(-0.828014\pi\)
							0.514402	+	0.857549i	\(0.328014\pi\)
\(17\)	65409.7			0.783153			0.391576	−	0.920146i	\(-0.371930\pi\)
							0.391576	+	0.920146i	\(0.371930\pi\)
\(19\)	−66925.9	−	66925.9i	−0.513546	−	0.513546i	0.402065	−	0.915611i	\(-0.368293\pi\)
							−0.915611	+	0.402065i	\(0.868293\pi\)
\(23\)	171177.			0.611695			0.305847	−	0.952081i	\(-0.401060\pi\)
							0.305847	+	0.952081i	\(0.401060\pi\)
\(29\)	−841274.	+	841274.i	−1.18945	+	1.18945i	−0.212229	+	0.977220i	\(0.568072\pi\)
							−0.977220	+	0.212229i	\(0.931928\pi\)
\(31\)		−	729280.i		−	0.789674i	−0.918751	−	0.394837i	\(-0.870801\pi\)
							0.918751	−	0.394837i	\(-0.129199\pi\)
\(37\)	−722145.	−	722145.i	−0.385316	−	0.385316i	0.487697	−	0.873013i	\(-0.337837\pi\)
							−0.873013	+	0.487697i	\(0.837837\pi\)
\(41\)			3.36723e6i			1.19162i	0.803125	+	0.595810i	\(0.203169\pi\)
							−0.803125	+	0.595810i	\(0.796831\pi\)
\(43\)	2.22438e6	−	2.22438e6i	0.650632	−	0.650632i	−0.302513	−	0.953145i	\(-0.597826\pi\)
							0.953145	+	0.302513i	\(0.0978256\pi\)
\(47\)			6.58869e6i			1.35023i	0.737713	+	0.675114i	\(0.235906\pi\)
							−0.737713	+	0.675114i	\(0.764094\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.9.l.a.43.2	yes	64
4.3	odd	2		192.9.l.a.79.32		64
16.3	odd	4	inner	48.9.l.a.19.2	✓	64
16.13	even	4		192.9.l.a.175.32		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.9.l.a.19.2	✓	64	16.3	odd	4	inner
48.9.l.a.43.2	yes	64	1.1	even	1	trivial
192.9.l.a.79.32		64	4.3	odd	2
192.9.l.a.175.32		64	16.13	even	4