Embedded newform 48.4.j.a.37.1

• Label: 48.4.j.a.37.1
• Level: $48$
• Weight: $4$
• Character: 48.37
• 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			37.1
Character	\(\chi\)	\(=\)	48.37
Dual form			48.4.j.a.13.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.77551 - 0.544550i) q^{2} +(-2.12132 + 2.12132i) q^{3} +(7.40693 + 3.02281i) q^{4} +(-3.72414 - 3.72414i) q^{5} +(7.04291 - 4.73259i) q^{6} -20.2675i q^{7} +(-18.9120 - 12.4233i) q^{8} -9.00000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 20 q^{4} + 84 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.77551	−	0.544550i	−0.981292	−	0.192527i
							\(3\)	−2.12132	+	2.12132i	−0.408248	+	0.408248i
\(5\)	−3.72414	−	3.72414i	−0.333097	−	0.333097i								0.520664	−	0.853762i	\(-0.325684\pi\)
							−0.853762	+	0.520664i	\(0.825684\pi\)
\(7\)		−	20.2675i		−	1.09434i	−0.837020	−	0.547172i	\(-0.815704\pi\)
							0.837020	−	0.547172i	\(-0.184296\pi\)
\(11\)	−47.5467	−	47.5467i	−1.30326	−	1.30326i	−0.926180	−	0.377081i	\(-0.876928\pi\)
							−0.377081	−	0.926180i	\(-0.623072\pi\)
\(13\)	27.8636	−	27.8636i	0.594460	−	0.594460i	−0.344373	−	0.938833i	\(-0.611909\pi\)
							0.938833	+	0.344373i	\(0.111909\pi\)
\(17\)	56.3788			0.804345			0.402173	−	0.915564i	\(-0.368255\pi\)
							0.402173	+	0.915564i	\(0.368255\pi\)
\(19\)	−66.1489	+	66.1489i	−0.798716	+	0.798716i	−0.982893	−	0.184177i	\(-0.941038\pi\)
							0.184177	+	0.982893i	\(0.441038\pi\)
\(23\)			3.66579i			0.0332335i	0.999862	+	0.0166167i	\(0.00528951\pi\)
							−0.999862	+	0.0166167i	\(0.994710\pi\)
\(29\)	−86.8428	+	86.8428i	−0.556079	+	0.556079i	−0.928189	−	0.372109i	\(-0.878635\pi\)
							0.372109	+	0.928189i	\(0.378635\pi\)
\(31\)	−102.846			−0.595858			−0.297929	−	0.954588i	\(-0.596296\pi\)
							−0.297929	+	0.954588i	\(0.596296\pi\)
\(37\)	66.8267	+	66.8267i	0.296925	+	0.296925i	0.839808	−	0.542883i	\(-0.182667\pi\)
							−0.542883	+	0.839808i	\(0.682667\pi\)
\(41\)		−	29.5734i		−	0.112649i	−0.998413	−	0.0563243i	\(-0.982062\pi\)
							0.998413	−	0.0563243i	\(-0.0179381\pi\)
\(43\)	−372.929	−	372.929i	−1.32258	−	1.32258i	−0.911678	−	0.410905i	\(-0.865213\pi\)
							−0.410905	−	0.911678i	\(-0.634787\pi\)
\(47\)	539.082			1.67305			0.836524	−	0.547930i	\(-0.184584\pi\)
							0.836524	+	0.547930i	\(0.184584\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.4.j.a.37.1	yes	24
3.2	odd	2		144.4.k.b.37.12		24
4.3	odd	2		192.4.j.a.49.9		24
8.3	odd	2		384.4.j.a.97.5		24
8.5	even	2		384.4.j.b.97.8		24
12.11	even	2		576.4.k.b.433.8		24
16.3	odd	4		192.4.j.a.145.9		24
16.5	even	4		384.4.j.b.289.8		24
16.11	odd	4		384.4.j.a.289.5		24
16.13	even	4	inner	48.4.j.a.13.1	✓	24
48.29	odd	4		144.4.k.b.109.12		24
48.35	even	4		576.4.k.b.145.8		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.4.j.a.13.1	✓	24	16.13	even	4	inner
48.4.j.a.37.1	yes	24	1.1	even	1	trivial
144.4.k.b.37.12		24	3.2	odd	2
144.4.k.b.109.12		24	48.29	odd	4
192.4.j.a.49.9		24	4.3	odd	2
192.4.j.a.145.9		24	16.3	odd	4
384.4.j.a.97.5		24	8.3	odd	2
384.4.j.a.289.5		24	16.11	odd	4
384.4.j.b.97.8		24	8.5	even	2
384.4.j.b.289.8		24	16.5	even	4
576.4.k.b.145.8		24	48.35	even	4
576.4.k.b.433.8		24	12.11	even	2