Embedded newform 48.3.l.a.19.4

• Label: 48.3.l.a.19.4
• 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			19.4
Root			\(0.125358 - 1.99607i\) of defining polynomial
Character	\(\chi\)	\(=\)	48.19
Dual form			48.3.l.a.43.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.125358 - 1.99607i) q^{2} +(1.22474 - 1.22474i) q^{3} +(-3.96857 + 0.500444i) q^{4} +(3.32679 - 3.32679i) q^{5} +(-2.59820 - 2.29114i) q^{6} -4.04088 q^{7} +(1.49641 + 7.85880i) q^{8} -3.00000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 12 q^{4} - 12 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.125358	−	1.99607i	−0.0626788	−	0.998034i
							\(3\)	1.22474	−	1.22474i	0.408248	−	0.408248i
\(5\)	3.32679	−	3.32679i	0.665359	−	0.665359i								−0.291279	−	0.956638i	\(-0.594081\pi\)
							0.956638	+	0.291279i	\(0.0940809\pi\)
\(7\)	−4.04088			−0.577269			−0.288635	−	0.957439i	\(-0.593201\pi\)
							−0.288635	+	0.957439i	\(0.593201\pi\)
\(11\)	6.82458	+	6.82458i	0.620416	+	0.620416i	0.945638	−	0.325222i	\(-0.105439\pi\)
							−0.325222	+	0.945638i	\(0.605439\pi\)
\(13\)	4.29091	+	4.29091i	0.330070	+	0.330070i	0.852613	−	0.522543i	\(-0.175017\pi\)
							−0.522543	+	0.852613i	\(0.675017\pi\)
\(17\)	30.1192			1.77172			0.885859	−	0.463954i	\(-0.153570\pi\)
							0.885859	+	0.463954i	\(0.153570\pi\)
\(19\)	−19.7548	+	19.7548i	−1.03973	+	1.03973i	−0.0405505	+	0.999177i	\(0.512911\pi\)
							−0.999177	+	0.0405505i	\(0.987089\pi\)
\(23\)	−28.2345			−1.22759			−0.613794	−	0.789466i	\(-0.710358\pi\)
							−0.613794	+	0.789466i	\(0.710358\pi\)
\(29\)	−21.3607	−	21.3607i	−0.736575	−	0.736575i	0.235338	−	0.971914i	\(-0.424380\pi\)
							−0.971914	+	0.235338i	\(0.924380\pi\)
\(31\)		−	38.0396i		−	1.22708i	−0.789662	−	0.613541i	\(-0.789744\pi\)
							0.789662	−	0.613541i	\(-0.210256\pi\)
\(37\)	−42.8916	+	42.8916i	−1.15923	+	1.15923i	−0.174590	+	0.984641i	\(0.555860\pi\)
							−0.984641	+	0.174590i	\(0.944140\pi\)
\(41\)		−	48.2343i		−	1.17645i	−0.808699	−	0.588223i	\(-0.799828\pi\)
							0.808699	−	0.588223i	\(-0.200172\pi\)
\(43\)	32.6765	+	32.6765i	0.759918	+	0.759918i	0.976307	−	0.216389i	\(-0.0694281\pi\)
							−0.216389	+	0.976307i	\(0.569428\pi\)
\(47\)			15.8305i			0.336818i	0.985717	+	0.168409i	\(0.0538630\pi\)
							−0.985717	+	0.168409i	\(0.946137\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.3.l.a.19.4	✓	16
3.2	odd	2		144.3.m.c.19.5		16
4.3	odd	2		192.3.l.a.175.4		16
8.3	odd	2		384.3.l.b.223.5		16
8.5	even	2		384.3.l.a.223.1		16
12.11	even	2		576.3.m.c.559.2		16
16.3	odd	4		384.3.l.a.31.1		16
16.5	even	4		192.3.l.a.79.4		16
16.11	odd	4	inner	48.3.l.a.43.4	yes	16
16.13	even	4		384.3.l.b.31.5		16
24.5	odd	2		1152.3.m.f.991.7		16
24.11	even	2		1152.3.m.c.991.7		16
48.5	odd	4		576.3.m.c.271.2		16
48.11	even	4		144.3.m.c.91.5		16
48.29	odd	4		1152.3.m.c.415.7		16
48.35	even	4		1152.3.m.f.415.7		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.3.l.a.19.4	✓	16	1.1	even	1	trivial
48.3.l.a.43.4	yes	16	16.11	odd	4	inner
144.3.m.c.19.5		16	3.2	odd	2
144.3.m.c.91.5		16	48.11	even	4
192.3.l.a.79.4		16	16.5	even	4
192.3.l.a.175.4		16	4.3	odd	2
384.3.l.a.31.1		16	16.3	odd	4
384.3.l.a.223.1		16	8.5	even	2
384.3.l.b.31.5		16	16.13	even	4
384.3.l.b.223.5		16	8.3	odd	2
576.3.m.c.271.2		16	48.5	odd	4
576.3.m.c.559.2		16	12.11	even	2
1152.3.m.c.415.7		16	48.29	odd	4
1152.3.m.c.991.7		16	24.11	even	2
1152.3.m.f.415.7		16	48.35	even	4
1152.3.m.f.991.7		16	24.5	odd	2