Embedded newform 48.3.l.a.19.8

• Label: 48.3.l.a.19.8
• 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			19.8
Root			\(-1.96679 + 0.362960i\) of defining polynomial
Character	\(\chi\)	\(=\)	48.19
Dual form			48.3.l.a.43.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.96679 + 0.362960i) q^{2} +(-1.22474 + 1.22474i) q^{3} +(3.73652 + 1.42773i) q^{4} +(1.69930 - 1.69930i) q^{5} +(-2.85335 + 1.96428i) q^{6} -5.74280 q^{7} +(6.83074 + 4.16426i) q^{8} -3.00000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 12 q^{4} - 12 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.96679	+	0.362960i	0.983395	+	0.181480i
							\(3\)	−1.22474	+	1.22474i	−0.408248	+	0.408248i
\(5\)	1.69930	−	1.69930i	0.339861	−	0.339861i								−0.516454	−	0.856315i	\(-0.672748\pi\)
							0.856315	+	0.516454i	\(0.172748\pi\)
\(7\)	−5.74280			−0.820400			−0.410200	−	0.911996i	\(-0.634541\pi\)
							−0.410200	+	0.911996i	\(0.634541\pi\)
\(11\)	−5.59560	−	5.59560i	−0.508691	−	0.508691i	0.405434	−	0.914125i	\(-0.367121\pi\)
							−0.914125	+	0.405434i	\(0.867121\pi\)
\(13\)	−13.5782	−	13.5782i	−1.04447	−	1.04447i	−0.998964	−	0.0455110i	\(-0.985508\pi\)
							−0.0455110	−	0.998964i	\(-0.514492\pi\)
\(17\)	19.7023			1.15896			0.579481	−	0.814986i	\(-0.303255\pi\)
							0.579481	+	0.814986i	\(0.303255\pi\)
\(19\)	−21.6943	+	21.6943i	−1.14181	+	1.14181i	−0.153687	+	0.988120i	\(0.549115\pi\)
							−0.988120	+	0.153687i	\(0.950885\pi\)
\(23\)	24.9257			1.08373			0.541863	−	0.840467i	\(-0.317719\pi\)
							0.541863	+	0.840467i	\(0.317719\pi\)
\(29\)	1.50581	+	1.50581i	0.0519245	+	0.0519245i	0.732592	−	0.680668i	\(-0.238310\pi\)
							−0.680668	+	0.732592i	\(0.738310\pi\)
\(31\)		−	2.20037i		−	0.0709796i	−0.999370	−	0.0354898i	\(-0.988701\pi\)
							0.999370	−	0.0354898i	\(-0.0112991\pi\)
\(37\)	27.6956	−	27.6956i	0.748530	−	0.748530i	−0.225673	−	0.974203i	\(-0.572458\pi\)
							0.974203	+	0.225673i	\(0.0724580\pi\)
\(41\)			51.3127i			1.25153i	0.780012	+	0.625764i	\(0.215213\pi\)
							−0.780012	+	0.625764i	\(0.784787\pi\)
\(43\)	21.4400	+	21.4400i	0.498606	+	0.498606i	0.911004	−	0.412398i	\(-0.135309\pi\)
							−0.412398	+	0.911004i	\(0.635309\pi\)
\(47\)		−	76.5216i		−	1.62812i	−0.580781	−	0.814060i	\(-0.697253\pi\)
							0.580781	−	0.814060i	\(-0.302747\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.3.l.a.19.8	✓	16
3.2	odd	2		144.3.m.c.19.1		16
4.3	odd	2		192.3.l.a.175.7		16
8.3	odd	2		384.3.l.b.223.2		16
8.5	even	2		384.3.l.a.223.6		16
12.11	even	2		576.3.m.c.559.3		16
16.3	odd	4		384.3.l.a.31.6		16
16.5	even	4		192.3.l.a.79.7		16
16.11	odd	4	inner	48.3.l.a.43.8	yes	16
16.13	even	4		384.3.l.b.31.2		16
24.5	odd	2		1152.3.m.f.991.6		16
24.11	even	2		1152.3.m.c.991.6		16
48.5	odd	4		576.3.m.c.271.3		16
48.11	even	4		144.3.m.c.91.1		16
48.29	odd	4		1152.3.m.c.415.6		16
48.35	even	4		1152.3.m.f.415.6		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.3.l.a.19.8	✓	16	1.1	even	1	trivial
48.3.l.a.43.8	yes	16	16.11	odd	4	inner
144.3.m.c.19.1		16	3.2	odd	2
144.3.m.c.91.1		16	48.11	even	4
192.3.l.a.79.7		16	16.5	even	4
192.3.l.a.175.7		16	4.3	odd	2
384.3.l.a.31.6		16	16.3	odd	4
384.3.l.a.223.6		16	8.5	even	2
384.3.l.b.31.2		16	16.13	even	4
384.3.l.b.223.2		16	8.3	odd	2
576.3.m.c.271.3		16	48.5	odd	4
576.3.m.c.559.3		16	12.11	even	2
1152.3.m.c.415.6		16	48.29	odd	4
1152.3.m.c.991.6		16	24.11	even	2
1152.3.m.f.415.6		16	48.35	even	4
1152.3.m.f.991.6		16	24.5	odd	2