Embedded newform 48.3.l.a.19.1

• Label: 48.3.l.a.19.1
• Level: $48$
• Weight: $3$
• Character: 48.19
• Analytic conductor: $1.308$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.30790526893\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	x^16 - 6*x^14 - 4*x^13 + 10*x^12 + 56*x^11 + 88*x^10 - 128*x^9 - 496*x^8 - 512*x^7 + 1408*x^6 + 3584*x^5 + 2560*x^4 - 4096*x^3 - 24576*x^2 + 65536
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{9} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			19.1
Root			\(1.84258 + 0.777752i\) of defining polynomial
Character	\(\chi\)	\(=\)	48.19
Dual form			48.3.l.a.43.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.84258 + 0.777752i) q^{2} +(-1.22474 + 1.22474i) q^{3} +(2.79020 - 2.86614i) q^{4} +(-4.78830 + 4.78830i) q^{5} +(1.30414 - 3.20924i) q^{6} -10.3302 q^{7} +(-2.91202 + 7.45118i) q^{8} -3.00000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 12 q^{4} - 12 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.84258	+	0.777752i	−0.921290	+	0.388876i
							\(3\)	−1.22474	+	1.22474i	−0.408248	+	0.408248i
\(5\)	−4.78830	+	4.78830i	−0.957661	+	0.957661i								−0.999139	−	0.0414785i	\(-0.986793\pi\)
							0.0414785	+	0.999139i	\(0.486793\pi\)
\(7\)	−10.3302			−1.47575			−0.737875	−	0.674937i	\(-0.764171\pi\)
							−0.737875	+	0.674937i	\(0.764171\pi\)
\(11\)	−0.526169	−	0.526169i	−0.0478335	−	0.0478335i	0.682785	−	0.730619i	\(-0.260768\pi\)
							−0.730619	+	0.682785i	\(0.760768\pi\)
\(13\)	17.2840	+	17.2840i	1.32953	+	1.32953i	0.905774	+	0.423761i	\(0.139290\pi\)
							0.423761	+	0.905774i	\(0.360710\pi\)
\(17\)	4.71650			0.277441			0.138721	−	0.990332i	\(-0.455701\pi\)
							0.138721	+	0.990332i	\(0.455701\pi\)
\(19\)	−2.53604	+	2.53604i	−0.133476	+	0.133476i	−0.770688	−	0.637213i	\(-0.780087\pi\)
							0.637213	+	0.770688i	\(0.280087\pi\)
\(23\)	−12.5864			−0.547236			−0.273618	−	0.961838i	\(-0.588220\pi\)
							−0.273618	+	0.961838i	\(0.588220\pi\)
\(29\)	−2.19683	−	2.19683i	−0.0757526	−	0.0757526i	0.668215	−	0.743968i	\(-0.267058\pi\)
							−0.743968	+	0.668215i	\(0.767058\pi\)
\(31\)			28.0521i			0.904908i	0.891788	+	0.452454i	\(0.149451\pi\)
							−0.891788	+	0.452454i	\(0.850549\pi\)
\(37\)	−32.1128	+	32.1128i	−0.867914	+	0.867914i	−0.992241	−	0.124327i	\(-0.960323\pi\)
							0.124327	+	0.992241i	\(0.460323\pi\)
\(41\)		−	23.1145i		−	0.563768i	−0.959449	−	0.281884i	\(-0.909041\pi\)
							0.959449	−	0.281884i	\(-0.0909593\pi\)
\(43\)	4.79441	+	4.79441i	0.111498	+	0.111498i	0.760655	−	0.649157i	\(-0.224878\pi\)
							−0.649157	+	0.760655i	\(0.724878\pi\)
\(47\)		−	39.0095i		−	0.829989i	−0.909824	−	0.414994i	\(-0.863784\pi\)
							0.909824	−	0.414994i	\(-0.136216\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.3.l.a.19.1	✓	16
3.2	odd	2		144.3.m.c.19.8		16
4.3	odd	2		192.3.l.a.175.5		16
8.3	odd	2		384.3.l.b.223.4		16
8.5	even	2		384.3.l.a.223.8		16
12.11	even	2		576.3.m.c.559.7		16
16.3	odd	4		384.3.l.a.31.8		16
16.5	even	4		192.3.l.a.79.5		16
16.11	odd	4	inner	48.3.l.a.43.1	yes	16
16.13	even	4		384.3.l.b.31.4		16
24.5	odd	2		1152.3.m.f.991.2		16
24.11	even	2		1152.3.m.c.991.2		16
48.5	odd	4		576.3.m.c.271.7		16
48.11	even	4		144.3.m.c.91.8		16
48.29	odd	4		1152.3.m.c.415.2		16
48.35	even	4		1152.3.m.f.415.2		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.3.l.a.19.1	✓	16	1.1	even	1	trivial
48.3.l.a.43.1	yes	16	16.11	odd	4	inner
144.3.m.c.19.8		16	3.2	odd	2
144.3.m.c.91.8		16	48.11	even	4
192.3.l.a.79.5		16	16.5	even	4
192.3.l.a.175.5		16	4.3	odd	2
384.3.l.a.31.8		16	16.3	odd	4
384.3.l.a.223.8		16	8.5	even	2
384.3.l.b.31.4		16	16.13	even	4
384.3.l.b.223.4		16	8.3	odd	2
576.3.m.c.271.7		16	48.5	odd	4
576.3.m.c.559.7		16	12.11	even	2
1152.3.m.c.415.2		16	48.29	odd	4
1152.3.m.c.991.2		16	24.11	even	2
1152.3.m.f.415.2		16	48.35	even	4
1152.3.m.f.991.2		16	24.5	odd	2