Embedded newform 48.3.l.a.19.5

• Label: 48.3.l.a.19.5
• 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			19.5
Root			\(-0.455024 + 1.94755i\) of defining polynomial
Character	\(\chi\)	\(=\)	48.19
Dual form			48.3.l.a.43.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.455024 + 1.94755i) q^{2} +(-1.22474 + 1.22474i) q^{3} +(-3.58591 + 1.77236i) q^{4} +(-3.40572 + 3.40572i) q^{5} +(-2.94254 - 1.82796i) q^{6} +12.1303 q^{7} +(-5.08344 - 6.17727i) q^{8} -3.00000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 12 q^{4} - 12 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.455024	+	1.94755i	0.227512	+	0.973775i
							\(3\)	−1.22474	+	1.22474i	−0.408248	+	0.408248i
\(5\)	−3.40572	+	3.40572i	−0.681145	+	0.681145i								−0.960258	−	0.279113i	\(-0.909960\pi\)
							0.279113	+	0.960258i	\(0.409960\pi\)
\(7\)	12.1303			1.73290			0.866452	−	0.499261i	\(-0.166395\pi\)
							0.866452	+	0.499261i	\(0.166395\pi\)
\(11\)	9.81086	+	9.81086i	0.891896	+	0.891896i	0.994702	−	0.102805i	\(-0.0327818\pi\)
							−0.102805	+	0.994702i	\(0.532782\pi\)
\(13\)	−7.76859	−	7.76859i	−0.597584	−	0.597584i	0.342085	−	0.939669i	\(-0.388867\pi\)
							−0.939669	+	0.342085i	\(0.888867\pi\)
\(17\)	9.73087			0.572404			0.286202	−	0.958169i	\(-0.407607\pi\)
							0.286202	+	0.958169i	\(0.407607\pi\)
\(19\)	11.2823	−	11.2823i	0.593806	−	0.593806i	−0.344851	−	0.938657i	\(-0.612071\pi\)
							0.938657	+	0.344851i	\(0.112071\pi\)
\(23\)	−20.2635			−0.881020			−0.440510	−	0.897748i	\(-0.645202\pi\)
							−0.440510	+	0.897748i	\(0.645202\pi\)
\(29\)	−16.4069	−	16.4069i	−0.565754	−	0.565754i	0.365182	−	0.930936i	\(-0.381007\pi\)
							−0.930936	+	0.365182i	\(0.881007\pi\)
\(31\)		−	26.3542i		−	0.850134i	−0.905162	−	0.425067i	\(-0.860251\pi\)
							0.905162	−	0.425067i	\(-0.139749\pi\)
\(37\)	−23.7263	+	23.7263i	−0.641250	+	0.641250i	−0.950863	−	0.309613i	\(-0.899801\pi\)
							0.309613	+	0.950863i	\(0.399801\pi\)
\(41\)		−	24.7452i		−	0.603542i	−0.953380	−	0.301771i	\(-0.902422\pi\)
							0.953380	−	0.301771i	\(-0.0975779\pi\)
\(43\)	29.8844	+	29.8844i	0.694987	+	0.694987i	0.963325	−	0.268338i	\(-0.0864744\pi\)
							−0.268338	+	0.963325i	\(0.586474\pi\)
\(47\)		−	31.3325i		−	0.666648i	−0.942812	−	0.333324i	\(-0.891830\pi\)
							0.942812	−	0.333324i	\(-0.108170\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.3.l.a.19.5	✓	16
3.2	odd	2		144.3.m.c.19.4		16
4.3	odd	2		192.3.l.a.175.6		16
8.3	odd	2		384.3.l.b.223.3		16
8.5	even	2		384.3.l.a.223.7		16
12.11	even	2		576.3.m.c.559.6		16
16.3	odd	4		384.3.l.a.31.7		16
16.5	even	4		192.3.l.a.79.6		16
16.11	odd	4	inner	48.3.l.a.43.5	yes	16
16.13	even	4		384.3.l.b.31.3		16
24.5	odd	2		1152.3.m.f.991.3		16
24.11	even	2		1152.3.m.c.991.3		16
48.5	odd	4		576.3.m.c.271.6		16
48.11	even	4		144.3.m.c.91.4		16
48.29	odd	4		1152.3.m.c.415.3		16
48.35	even	4		1152.3.m.f.415.3		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.3.l.a.19.5	✓	16	1.1	even	1	trivial
48.3.l.a.43.5	yes	16	16.11	odd	4	inner
144.3.m.c.19.4		16	3.2	odd	2
144.3.m.c.91.4		16	48.11	even	4
192.3.l.a.79.6		16	16.5	even	4
192.3.l.a.175.6		16	4.3	odd	2
384.3.l.a.31.7		16	16.3	odd	4
384.3.l.a.223.7		16	8.5	even	2
384.3.l.b.31.3		16	16.13	even	4
384.3.l.b.223.3		16	8.3	odd	2
576.3.m.c.271.6		16	48.5	odd	4
576.3.m.c.559.6		16	12.11	even	2
1152.3.m.c.415.3		16	48.29	odd	4
1152.3.m.c.991.3		16	24.11	even	2
1152.3.m.f.415.3		16	48.35	even	4
1152.3.m.f.991.3		16	24.5	odd	2