Embedded newform 48.3.l.a.19.7

• Label: 48.3.l.a.19.7
• 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			19.7
Root			\(-1.87459 - 0.697079i\) of defining polynomial
Character	\(\chi\)	\(=\)	48.19
Dual form			48.3.l.a.43.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.87459 - 0.697079i) q^{2} +(1.22474 - 1.22474i) q^{3} +(3.02816 - 2.61347i) q^{4} +(-5.24354 + 5.24354i) q^{5} +(1.44215 - 3.14964i) q^{6} -5.32796 q^{7} +(3.85476 - 7.01005i) q^{8} -3.00000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 12 q^{4} - 12 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.87459	−	0.697079i	0.937294	−	0.348540i
							\(3\)	1.22474	−	1.22474i	0.408248	−	0.408248i
\(5\)	−5.24354	+	5.24354i	−1.04871	+	1.04871i								−0.0499563	+	0.998751i	\(0.515908\pi\)
							−0.998751	+	0.0499563i	\(0.984092\pi\)
\(7\)	−5.32796			−0.761138			−0.380569	−	0.924753i	\(-0.624272\pi\)
							−0.380569	+	0.924753i	\(0.624272\pi\)
\(11\)	12.2863	+	12.2863i	1.11693	+	1.11693i	0.992189	+	0.124743i	\(0.0398107\pi\)
							0.124743	+	0.992189i	\(0.460189\pi\)
\(13\)	−5.73657	−	5.73657i	−0.441275	−	0.441275i	0.451165	−	0.892440i	\(-0.351008\pi\)
							−0.892440	+	0.451165i	\(0.851008\pi\)
\(17\)	−23.3997			−1.37645			−0.688226	−	0.725496i	\(-0.741610\pi\)
							−0.688226	+	0.725496i	\(0.741610\pi\)
\(19\)	11.7492	−	11.7492i	0.618380	−	0.618380i	−0.326736	−	0.945116i	\(-0.605949\pi\)
							0.945116	+	0.326736i	\(0.105949\pi\)
\(23\)	5.80841			0.252540			0.126270	−	0.991996i	\(-0.459699\pi\)
							0.126270	+	0.991996i	\(0.459699\pi\)
\(29\)	18.3914	+	18.3914i	0.634185	+	0.634185i	0.949115	−	0.314930i	\(-0.101981\pi\)
							−0.314930	+	0.949115i	\(0.601981\pi\)
\(31\)		−	16.9053i		−	0.545332i	−0.962109	−	0.272666i	\(-0.912095\pi\)
							0.962109	−	0.272666i	\(-0.0879053\pi\)
\(37\)	15.3391	−	15.3391i	0.414571	−	0.414571i	−0.468756	−	0.883327i	\(-0.655298\pi\)
							0.883327	+	0.468756i	\(0.155298\pi\)
\(41\)		−	29.2351i		−	0.713051i	−0.934286	−	0.356526i	\(-0.883961\pi\)
							0.934286	−	0.356526i	\(-0.116039\pi\)
\(43\)	33.4099	+	33.4099i	0.776975	+	0.776975i	0.979315	−	0.202340i	\(-0.0648546\pi\)
							−0.202340	+	0.979315i	\(0.564855\pi\)
\(47\)		−	18.2125i		−	0.387500i	−0.981051	−	0.193750i	\(-0.937935\pi\)
							0.981051	−	0.193750i	\(-0.0620650\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.3.l.a.19.7	✓	16
3.2	odd	2		144.3.m.c.19.2		16
4.3	odd	2		192.3.l.a.175.1		16
8.3	odd	2		384.3.l.b.223.8		16
8.5	even	2		384.3.l.a.223.4		16
12.11	even	2		576.3.m.c.559.8		16
16.3	odd	4		384.3.l.a.31.4		16
16.5	even	4		192.3.l.a.79.1		16
16.11	odd	4	inner	48.3.l.a.43.7	yes	16
16.13	even	4		384.3.l.b.31.8		16
24.5	odd	2		1152.3.m.f.991.1		16
24.11	even	2		1152.3.m.c.991.1		16
48.5	odd	4		576.3.m.c.271.8		16
48.11	even	4		144.3.m.c.91.2		16
48.29	odd	4		1152.3.m.c.415.1		16
48.35	even	4		1152.3.m.f.415.1		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.3.l.a.19.7	✓	16	1.1	even	1	trivial
48.3.l.a.43.7	yes	16	16.11	odd	4	inner
144.3.m.c.19.2		16	3.2	odd	2
144.3.m.c.91.2		16	48.11	even	4
192.3.l.a.79.1		16	16.5	even	4
192.3.l.a.175.1		16	4.3	odd	2
384.3.l.a.31.4		16	16.3	odd	4
384.3.l.a.223.4		16	8.5	even	2
384.3.l.b.31.8		16	16.13	even	4
384.3.l.b.223.8		16	8.3	odd	2
576.3.m.c.271.8		16	48.5	odd	4
576.3.m.c.559.8		16	12.11	even	2
1152.3.m.c.415.1		16	48.29	odd	4
1152.3.m.c.991.1		16	24.11	even	2
1152.3.m.f.415.1		16	48.35	even	4
1152.3.m.f.991.1		16	24.5	odd	2