Embedded newform 48.3.i.b.5.5

• Label: 48.3.i.b.5.5
• Level: $48$
• Weight: $3$
• Character: 48.5
• Analytic conductor: $1.308$
• Analytic rank: $0$
• Dimension: $20$
• 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:	\(20\)
Relative dimension:	\(10\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial:	x^20 - 2*x^18 + 6*x^16 - 24*x^14 - 24*x^12 + 1216*x^10 - 384*x^8 - 6144*x^6 + 24576*x^4 - 131072*x^2 + 1048576
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{13} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			5.5
Root			\(-0.312316 + 1.97546i\) of defining polynomial
Character	\(\chi\)	\(=\)	48.5
Dual form			48.3.i.b.29.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.312316 - 1.97546i) q^{2} +(-1.18505 - 2.75602i) q^{3} +(-3.80492 + 1.23394i) q^{4} +(-0.00985921 + 0.00985921i) q^{5} +(-5.07432 + 3.20176i) q^{6} -6.42277i q^{7} +(3.62594 + 7.13110i) q^{8} +(-6.19134 + 6.53203i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 6 q^{3} + 4 q^{4} - 12 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.312316	−	1.97546i	−0.156158	−	0.987732i
							\(3\)	−1.18505	−	2.75602i	−0.395015	−	0.918675i
\(5\)	−0.00985921	+	0.00985921i	−0.00197184	+	0.00197184i								−0.708092	−	0.706120i	\(-0.750444\pi\)
							0.706120	+	0.708092i	\(0.250444\pi\)
\(7\)		−	6.42277i		−	0.917538i	−0.888556	−	0.458769i	\(-0.848291\pi\)
							0.888556	−	0.458769i	\(-0.151709\pi\)
\(11\)	9.07186	−	9.07186i	0.824715	−	0.824715i	−0.162065	−	0.986780i	\(-0.551815\pi\)
							0.986780	+	0.162065i	\(0.0518155\pi\)
\(13\)	12.6098	−	12.6098i	0.969987	−	0.969987i	−0.0295753	−	0.999563i	\(-0.509415\pi\)
							0.999563	+	0.0295753i	\(0.00941548\pi\)
\(17\)			19.0155i			1.11856i	0.828978	+	0.559281i	\(0.188923\pi\)
							−0.828978	+	0.559281i	\(0.811077\pi\)
\(19\)	−2.07165	+	2.07165i	−0.109034	+	0.109034i	−0.759519	−	0.650485i	\(-0.774566\pi\)
							0.650485	+	0.759519i	\(0.274566\pi\)
\(23\)	19.5712			0.850923			0.425461	−	0.904977i	\(-0.360112\pi\)
							0.425461	+	0.904977i	\(0.360112\pi\)
\(29\)	−11.1742	−	11.1742i	−0.385319	−	0.385319i	0.487695	−	0.873014i	\(-0.337838\pi\)
							−0.873014	+	0.487695i	\(0.837838\pi\)
\(31\)	−59.9385			−1.93350			−0.966750	−	0.255725i	\(-0.917686\pi\)
							−0.966750	+	0.255725i	\(0.917686\pi\)
\(37\)	9.32707	+	9.32707i	0.252083	+	0.252083i	0.821824	−	0.569741i	\(-0.192957\pi\)
							−0.569741	+	0.821824i	\(0.692957\pi\)
\(41\)	47.2639			1.15278			0.576389	−	0.817176i	\(-0.304461\pi\)
							0.576389	+	0.817176i	\(0.304461\pi\)
\(43\)	24.1220	+	24.1220i	0.560978	+	0.560978i	0.929585	−	0.368607i	\(-0.120165\pi\)
							−0.368607	+	0.929585i	\(0.620165\pi\)
\(47\)			6.29702i			0.133979i	0.997754	+	0.0669896i	\(0.0213394\pi\)
							−0.997754	+	0.0669896i	\(0.978661\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.3.i.b.5.5	✓	20
3.2	odd	2	inner	48.3.i.b.5.6	yes	20
4.3	odd	2		192.3.i.b.113.6		20
8.3	odd	2		384.3.i.c.353.5		20
8.5	even	2		384.3.i.d.353.6		20
12.11	even	2		192.3.i.b.113.2		20
16.3	odd	4		192.3.i.b.17.2		20
16.5	even	4		384.3.i.d.161.2		20
16.11	odd	4		384.3.i.c.161.9		20
16.13	even	4	inner	48.3.i.b.29.6	yes	20
24.5	odd	2		384.3.i.d.353.2		20
24.11	even	2		384.3.i.c.353.9		20
48.5	odd	4		384.3.i.d.161.6		20
48.11	even	4		384.3.i.c.161.5		20
48.29	odd	4	inner	48.3.i.b.29.5	yes	20
48.35	even	4		192.3.i.b.17.6		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.3.i.b.5.5	✓	20	1.1	even	1	trivial
48.3.i.b.5.6	yes	20	3.2	odd	2	inner
48.3.i.b.29.5	yes	20	48.29	odd	4	inner
48.3.i.b.29.6	yes	20	16.13	even	4	inner
192.3.i.b.17.2		20	16.3	odd	4
192.3.i.b.17.6		20	48.35	even	4
192.3.i.b.113.2		20	12.11	even	2
192.3.i.b.113.6		20	4.3	odd	2
384.3.i.c.161.5		20	48.11	even	4
384.3.i.c.161.9		20	16.11	odd	4
384.3.i.c.353.5		20	8.3	odd	2
384.3.i.c.353.9		20	24.11	even	2
384.3.i.d.161.2		20	16.5	even	4
384.3.i.d.161.6		20	48.5	odd	4
384.3.i.d.353.2		20	24.5	odd	2
384.3.i.d.353.6		20	8.5	even	2