Embedded newform 48.3.i.a.5.3

• Label: 48.3.i.a.5.3
• Level: $48$
• Weight: $3$
• Character: 48.5
• Analytic conductor: $1.308$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.30790526893\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(i)\)
Coefficient field:	8.0.629407744.1
Defining polynomial:	\( x^{8} - 2x^{6} + 2x^{4} - 8x^{2} + 16 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{3} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			5.3
Root			\(-0.767178 - 1.18804i\) of defining polynomial
Character	\(\chi\)	\(=\)	48.5
Dual form			48.3.i.a.29.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.420861 - 1.95522i) q^{2} +(1.13234 + 2.77809i) q^{3} +(-3.64575 - 1.64575i) q^{4} +(6.28651 - 6.28651i) q^{5} +(5.90834 - 1.04478i) q^{6} +1.64575i q^{7} +(-4.75216 + 6.43560i) q^{8} +(-6.43560 + 6.29150i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} - 8 q^{4} + 16 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.420861	−	1.95522i	0.210431	−	0.977609i
							\(3\)	1.13234	+	2.77809i	0.377447	+	0.926031i
\(5\)	6.28651	−	6.28651i	1.25730	−	1.25730i								0.304927	−	0.952376i	\(-0.401368\pi\)
							0.952376	−	0.304927i	\(-0.0986321\pi\)
\(7\)			1.64575i			0.235107i	0.993067	+	0.117554i	\(0.0375052\pi\)
							−0.993067	+	0.117554i	\(0.962495\pi\)
\(11\)	−4.75216	+	4.75216i	−0.432014	+	0.432014i	−0.889313	−	0.457299i	\(-0.848817\pi\)
							0.457299	+	0.889313i	\(0.348817\pi\)
\(13\)	−9.35425	+	9.35425i	−0.719558	+	0.719558i	−0.968515	−	0.248957i	\(-0.919912\pi\)
							0.248957	+	0.968515i	\(0.419912\pi\)
\(17\)		−	11.4859i		−	0.675644i	−0.941210	−	0.337822i	\(-0.890310\pi\)
							0.941210	−	0.337822i	\(-0.109690\pi\)
\(19\)	−8.58301	+	8.58301i	−0.451737	+	0.451737i	−0.895931	−	0.444194i	\(-0.853490\pi\)
							0.444194	+	0.895931i	\(0.353490\pi\)
\(23\)	16.2381			0.706004			0.353002	−	0.935623i	\(-0.385161\pi\)
							0.353002	+	0.935623i	\(0.385161\pi\)
\(29\)	−10.7405	−	10.7405i	−0.370362	−	0.370362i	0.497247	−	0.867609i	\(-0.334344\pi\)
							−0.867609	+	0.497247i	\(0.834344\pi\)
\(31\)	6.35425			0.204976			0.102488	−	0.994734i	\(-0.467320\pi\)
							0.102488	+	0.994734i	\(0.467320\pi\)
\(37\)	27.2288	+	27.2288i	0.735912	+	0.735912i	0.971784	−	0.235872i	\(-0.0757946\pi\)
							−0.235872	+	0.971784i	\(0.575795\pi\)
\(41\)	1.98162			0.0483323			0.0241662	−	0.999708i	\(-0.492307\pi\)
							0.0241662	+	0.999708i	\(0.492307\pi\)
\(43\)	−19.4170	−	19.4170i	−0.451558	−	0.451558i	0.444313	−	0.895871i	\(-0.353448\pi\)
							−0.895871	+	0.444313i	\(0.853448\pi\)
\(47\)		−	74.9474i		−	1.59463i	−0.603566	−	0.797313i	\(-0.706254\pi\)
							0.603566	−	0.797313i	\(-0.293746\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.3.i.a.5.3	yes	8
3.2	odd	2	inner	48.3.i.a.5.2	✓	8
4.3	odd	2		192.3.i.a.113.2		8
8.3	odd	2		384.3.i.b.353.3		8
8.5	even	2		384.3.i.a.353.2		8
12.11	even	2		192.3.i.a.113.4		8
16.3	odd	4		192.3.i.a.17.4		8
16.5	even	4		384.3.i.a.161.4		8
16.11	odd	4		384.3.i.b.161.1		8
16.13	even	4	inner	48.3.i.a.29.2	yes	8
24.5	odd	2		384.3.i.a.353.4		8
24.11	even	2		384.3.i.b.353.1		8
48.5	odd	4		384.3.i.a.161.2		8
48.11	even	4		384.3.i.b.161.3		8
48.29	odd	4	inner	48.3.i.a.29.3	yes	8
48.35	even	4		192.3.i.a.17.2		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.3.i.a.5.2	✓	8	3.2	odd	2	inner
48.3.i.a.5.3	yes	8	1.1	even	1	trivial
48.3.i.a.29.2	yes	8	16.13	even	4	inner
48.3.i.a.29.3	yes	8	48.29	odd	4	inner
192.3.i.a.17.2		8	48.35	even	4
192.3.i.a.17.4		8	16.3	odd	4
192.3.i.a.113.2		8	4.3	odd	2
192.3.i.a.113.4		8	12.11	even	2
384.3.i.a.161.2		8	48.5	odd	4
384.3.i.a.161.4		8	16.5	even	4
384.3.i.a.353.2		8	8.5	even	2
384.3.i.a.353.4		8	24.5	odd	2
384.3.i.b.161.1		8	16.11	odd	4
384.3.i.b.161.3		8	48.11	even	4
384.3.i.b.353.1		8	24.11	even	2
384.3.i.b.353.3		8	8.3	odd	2