Embedded newform 48.3.i.a.5.4

• Label: 48.3.i.a.5.4
• 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			5.4
Root			\(1.38255 - 0.297594i\) of defining polynomial
Character	\(\chi\)	\(=\)	48.5
Dual form			48.3.i.a.29.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.68014 + 1.08495i) q^{2} +(0.737922 - 2.90783i) q^{3} +(1.64575 + 3.64575i) q^{4} +(-1.57472 + 1.57472i) q^{5} +(4.39467 - 4.08495i) q^{6} -3.64575i q^{7} +(-1.19038 + 7.91094i) q^{8} +(-7.91094 - 4.29150i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} - 8 q^{4} + 16 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.68014	+	1.08495i	0.840071	+	0.542477i
							\(3\)	0.737922	−	2.90783i	0.245974	−	0.969276i
\(5\)	−1.57472	+	1.57472i	−0.314944	+	0.314944i								−0.846821	−	0.531877i	\(-0.821487\pi\)
							0.531877	+	0.846821i	\(0.321487\pi\)
\(7\)		−	3.64575i		−	0.520822i	−0.965498	−	0.260411i	\(-0.916142\pi\)
							0.965498	−	0.260411i	\(-0.0838580\pi\)
\(11\)	−1.19038	+	1.19038i	−0.108216	+	0.108216i	−0.759142	−	0.650926i	\(-0.774381\pi\)
							0.650926	+	0.759142i	\(0.274381\pi\)
\(13\)	−14.6458	+	14.6458i	−1.12660	+	1.12660i	−0.135870	+	0.990727i	\(0.543383\pi\)
							−0.990727	+	0.135870i	\(0.956617\pi\)
\(17\)		−	28.0726i		−	1.65133i	−0.564159	−	0.825666i	\(-0.690800\pi\)
							0.564159	−	0.825666i	\(-0.309200\pi\)
\(19\)	12.5830	−	12.5830i	0.662263	−	0.662263i	−0.293650	−	0.955913i	\(-0.594870\pi\)
							0.955913	+	0.293650i	\(0.0948699\pi\)
\(23\)	29.2630			1.27231			0.636153	−	0.771563i	\(-0.280525\pi\)
							0.636153	+	0.771563i	\(0.280525\pi\)
\(29\)	19.3557	+	19.3557i	0.667437	+	0.667437i	0.957122	−	0.289685i	\(-0.0935506\pi\)
							−0.289685	+	0.957122i	\(0.593551\pi\)
\(31\)	11.6458			0.375669			0.187835	−	0.982201i	\(-0.439853\pi\)
							0.187835	+	0.982201i	\(0.439853\pi\)
\(37\)	0.771243	+	0.771243i	0.0208444	+	0.0208444i	0.717452	−	0.696608i	\(-0.245308\pi\)
							−0.696608	+	0.717452i	\(0.745308\pi\)
\(41\)	25.6919			0.626631			0.313316	−	0.949649i	\(-0.398560\pi\)
							0.313316	+	0.949649i	\(0.398560\pi\)
\(43\)	−40.5830	−	40.5830i	−0.943791	−	0.943791i	0.0547114	−	0.998502i	\(-0.482576\pi\)
							−0.998502	+	0.0547114i	\(0.982576\pi\)
\(47\)		−	50.2681i		−	1.06953i	−0.845000	−	0.534767i	\(-0.820399\pi\)
							0.845000	−	0.534767i	\(-0.179601\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.3.i.a.5.4	yes	8
3.2	odd	2	inner	48.3.i.a.5.1	✓	8
4.3	odd	2		192.3.i.a.113.3		8
8.3	odd	2		384.3.i.b.353.2		8
8.5	even	2		384.3.i.a.353.3		8
12.11	even	2		192.3.i.a.113.1		8
16.3	odd	4		192.3.i.a.17.1		8
16.5	even	4		384.3.i.a.161.1		8
16.11	odd	4		384.3.i.b.161.4		8
16.13	even	4	inner	48.3.i.a.29.1	yes	8
24.5	odd	2		384.3.i.a.353.1		8
24.11	even	2		384.3.i.b.353.4		8
48.5	odd	4		384.3.i.a.161.3		8
48.11	even	4		384.3.i.b.161.2		8
48.29	odd	4	inner	48.3.i.a.29.4	yes	8
48.35	even	4		192.3.i.a.17.3		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.3.i.a.5.1	✓	8	3.2	odd	2	inner
48.3.i.a.5.4	yes	8	1.1	even	1	trivial
48.3.i.a.29.1	yes	8	16.13	even	4	inner
48.3.i.a.29.4	yes	8	48.29	odd	4	inner
192.3.i.a.17.1		8	16.3	odd	4
192.3.i.a.17.3		8	48.35	even	4
192.3.i.a.113.1		8	12.11	even	2
192.3.i.a.113.3		8	4.3	odd	2
384.3.i.a.161.1		8	16.5	even	4
384.3.i.a.161.3		8	48.5	odd	4
384.3.i.a.353.1		8	24.5	odd	2
384.3.i.a.353.3		8	8.5	even	2
384.3.i.b.161.2		8	48.11	even	4
384.3.i.b.161.4		8	16.11	odd	4
384.3.i.b.353.2		8	8.3	odd	2
384.3.i.b.353.4		8	24.11	even	2