Embedded newform 48.4.j.a.13.3

• Label: 48.4.j.a.13.3
• Level: $48$
• Weight: $4$
• Character: 48.13
• Analytic conductor: $2.832$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.83209168028\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			13.3
Character	\(\chi\)	\(=\)	48.13
Dual form			48.4.j.a.37.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.92738 - 2.07008i) q^{2} +(2.12132 + 2.12132i) q^{3} +(-0.570442 + 7.97964i) q^{4} +(-7.29121 + 7.29121i) q^{5} +(0.302715 - 8.47988i) q^{6} +22.1610i q^{7} +(17.6179 - 14.1989i) q^{8} +9.00000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 20 q^{4} + 84 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.92738	−	2.07008i	−0.681430	−	0.731883i
							\(3\)	2.12132	+	2.12132i	0.408248	+	0.408248i
\(5\)	−7.29121	+	7.29121i	−0.652145	+	0.652145i								−0.953509	−	0.301364i	\(-0.902558\pi\)
							0.301364	+	0.953509i	\(0.402558\pi\)
\(7\)			22.1610i			1.19658i	0.801278	+	0.598292i	\(0.204154\pi\)
							−0.801278	+	0.598292i	\(0.795846\pi\)
\(11\)	8.24116	−	8.24116i	0.225891	−	0.225891i	−0.585083	−	0.810974i	\(-0.698938\pi\)
							0.810974	+	0.585083i	\(0.198938\pi\)
\(13\)	51.9094	+	51.9094i	1.10747	+	1.10747i	0.993482	+	0.113986i	\(0.0363618\pi\)
							0.113986	+	0.993482i	\(0.463638\pi\)
\(17\)	−58.7304			−0.837894			−0.418947	−	0.908011i	\(-0.637601\pi\)
							−0.418947	+	0.908011i	\(0.637601\pi\)
\(19\)	−54.5389	−	54.5389i	−0.658531	−	0.658531i	0.296501	−	0.955032i	\(-0.404180\pi\)
							−0.955032	+	0.296501i	\(0.904180\pi\)
\(23\)		−	117.989i		−	1.06967i	−0.844955	−	0.534837i	\(-0.820373\pi\)
							0.844955	−	0.534837i	\(-0.179627\pi\)
\(29\)	175.283	+	175.283i	1.12239	+	1.12239i	0.991381	+	0.131007i	\(0.0418212\pi\)
							0.131007	+	0.991381i	\(0.458179\pi\)
\(31\)	−6.58699			−0.0381631			−0.0190816	−	0.999818i	\(-0.506074\pi\)
							−0.0190816	+	0.999818i	\(0.506074\pi\)
\(37\)	265.451	−	265.451i	1.17946	−	1.17946i	0.199574	−	0.979883i	\(-0.436044\pi\)
							0.979883	−	0.199574i	\(-0.0639557\pi\)
\(41\)			98.7545i			0.376167i	0.982153	+	0.188084i	\(0.0602276\pi\)
							−0.982153	+	0.188084i	\(0.939772\pi\)
\(43\)	347.544	−	347.544i	1.23256	−	1.23256i	0.269580	−	0.962978i	\(-0.413115\pi\)
							0.962978	−	0.269580i	\(-0.0868850\pi\)
\(47\)	−141.880			−0.440327			−0.220164	−	0.975463i	\(-0.570659\pi\)
							−0.220164	+	0.975463i	\(0.570659\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.4.j.a.13.3	✓	24
3.2	odd	2		144.4.k.b.109.10		24
4.3	odd	2		192.4.j.a.145.2		24
8.3	odd	2		384.4.j.a.289.8		24
8.5	even	2		384.4.j.b.289.5		24
12.11	even	2		576.4.k.b.145.9		24
16.3	odd	4		384.4.j.a.97.8		24
16.5	even	4	inner	48.4.j.a.37.3	yes	24
16.11	odd	4		192.4.j.a.49.2		24
16.13	even	4		384.4.j.b.97.5		24
48.5	odd	4		144.4.k.b.37.10		24
48.11	even	4		576.4.k.b.433.9		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.4.j.a.13.3	✓	24	1.1	even	1	trivial
48.4.j.a.37.3	yes	24	16.5	even	4	inner
144.4.k.b.37.10		24	48.5	odd	4
144.4.k.b.109.10		24	3.2	odd	2
192.4.j.a.49.2		24	16.11	odd	4
192.4.j.a.145.2		24	4.3	odd	2
384.4.j.a.97.8		24	16.3	odd	4
384.4.j.a.289.8		24	8.3	odd	2
384.4.j.b.97.5		24	16.13	even	4
384.4.j.b.289.5		24	8.5	even	2
576.4.k.b.145.9		24	12.11	even	2
576.4.k.b.433.9		24	48.11	even	4