Embedded newform 48.3.i.b.5.2

• Label: 48.3.i.b.5.2
• Level: $48$
• Weight: $3$
• Character: 48.5
• Analytic conductor: $1.308$
• Analytic rank: $0$
• Dimension: $20$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.30790526893\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(10\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial:	x^20 - 2*x^18 + 6*x^16 - 24*x^14 - 24*x^12 + 1216*x^10 - 384*x^8 - 6144*x^6 + 24576*x^4 - 131072*x^2 + 1048576
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{13} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			5.2
Root			\(-1.85381 - 0.750590i\) of defining polynomial
Character	\(\chi\)	\(=\)	48.5
Dual form			48.3.i.b.29.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.85381 + 0.750590i) q^{2} +(1.50491 + 2.59524i) q^{3} +(2.87323 - 2.78290i) q^{4} +(-2.59897 + 2.59897i) q^{5} +(-4.73777 - 3.68151i) q^{6} +7.30027i q^{7} +(-3.23761 + 7.31559i) q^{8} +(-4.47050 + 7.81118i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 6 q^{3} + 4 q^{4} - 12 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\)	\(17\)	\(31\)	\(37\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.85381	+	0.750590i	−0.926906	+	0.375295i
							\(3\)	1.50491	+	2.59524i	0.501636	+	0.865079i
\(5\)	−2.59897	+	2.59897i	−0.519793	+	0.519793i								−0.917509	−	0.397716i	\(-0.869803\pi\)
							0.397716	+	0.917509i	\(0.369803\pi\)
\(7\)			7.30027i			1.04290i	0.853283	+	0.521448i	\(0.174608\pi\)
							−0.853283	+	0.521448i	\(0.825392\pi\)
\(11\)	11.3161	−	11.3161i	1.02873	−	1.02873i	0.0291601	−	0.999575i	\(-0.490717\pi\)
							0.999575	−	0.0291601i	\(-0.00928328\pi\)
\(13\)	−0.746462	+	0.746462i	−0.0574201	+	0.0574201i	−0.735234	−	0.677814i	\(-0.762928\pi\)
							0.677814	+	0.735234i	\(0.262928\pi\)
\(17\)		−	6.67452i		−	0.392619i	−0.980542	−	0.196310i	\(-0.937104\pi\)
							0.980542	−	0.196310i	\(-0.0628957\pi\)
\(19\)	22.1936	−	22.1936i	1.16809	−	1.16809i	0.185428	−	0.982658i	\(-0.440633\pi\)
							0.982658	−	0.185428i	\(-0.0593670\pi\)
\(23\)	21.4389			0.932128			0.466064	−	0.884751i	\(-0.345672\pi\)
							0.466064	+	0.884751i	\(0.345672\pi\)
\(29\)	1.54272	+	1.54272i	0.0531973	+	0.0531973i	0.733205	−	0.680008i	\(-0.238024\pi\)
							−0.680008	+	0.733205i	\(0.738024\pi\)
\(31\)	−14.6082			−0.471233			−0.235616	−	0.971846i	\(-0.575711\pi\)
							−0.235616	+	0.971846i	\(0.575711\pi\)
\(37\)	−50.1010	−	50.1010i	−1.35408	−	1.35408i	−0.881041	−	0.473039i	\(-0.843157\pi\)
							−0.473039	−	0.881041i	\(-0.656843\pi\)
\(41\)	−15.0731			−0.367637			−0.183819	−	0.982960i	\(-0.558846\pi\)
							−0.183819	+	0.982960i	\(0.558846\pi\)
\(43\)	26.3634	+	26.3634i	0.613102	+	0.613102i	0.943753	−	0.330651i	\(-0.107268\pi\)
							−0.330651	+	0.943753i	\(0.607268\pi\)
\(47\)			36.6067i			0.778866i	0.921055	+	0.389433i	\(0.127329\pi\)
							−0.921055	+	0.389433i	\(0.872671\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.3.i.b.5.2	✓	20
3.2	odd	2	inner	48.3.i.b.5.9	yes	20
4.3	odd	2		192.3.i.b.113.3		20
8.3	odd	2		384.3.i.c.353.8		20
8.5	even	2		384.3.i.d.353.3		20
12.11	even	2		192.3.i.b.113.9		20
16.3	odd	4		192.3.i.b.17.9		20
16.5	even	4		384.3.i.d.161.9		20
16.11	odd	4		384.3.i.c.161.2		20
16.13	even	4	inner	48.3.i.b.29.9	yes	20
24.5	odd	2		384.3.i.d.353.9		20
24.11	even	2		384.3.i.c.353.2		20
48.5	odd	4		384.3.i.d.161.3		20
48.11	even	4		384.3.i.c.161.8		20
48.29	odd	4	inner	48.3.i.b.29.2	yes	20
48.35	even	4		192.3.i.b.17.3		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.3.i.b.5.2	✓	20	1.1	even	1	trivial
48.3.i.b.5.9	yes	20	3.2	odd	2	inner
48.3.i.b.29.2	yes	20	48.29	odd	4	inner
48.3.i.b.29.9	yes	20	16.13	even	4	inner
192.3.i.b.17.3		20	48.35	even	4
192.3.i.b.17.9		20	16.3	odd	4
192.3.i.b.113.3		20	4.3	odd	2
192.3.i.b.113.9		20	12.11	even	2
384.3.i.c.161.2		20	16.11	odd	4
384.3.i.c.161.8		20	48.11	even	4
384.3.i.c.353.2		20	24.11	even	2
384.3.i.c.353.8		20	8.3	odd	2
384.3.i.d.161.3		20	48.5	odd	4
384.3.i.d.161.9		20	16.5	even	4
384.3.i.d.353.3		20	8.5	even	2
384.3.i.d.353.9		20	24.5	odd	2