Embedded newform 48.3.i.b.5.10

• Label: 48.3.i.b.5.10
• 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			5.10
Root			\(1.96139 + 0.391068i\) of defining polynomial
Character	\(\chi\)	\(=\)	48.5
Dual form			48.3.i.b.29.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.96139 - 0.391068i) q^{2} +(-0.164573 + 2.99548i) q^{3} +(3.69413 - 1.53408i) q^{4} +(-3.61305 + 3.61305i) q^{5} +(0.848646 + 5.93968i) q^{6} -12.2792i q^{7} +(6.64572 - 4.45358i) q^{8} +(-8.94583 - 0.985948i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 6 q^{3} + 4 q^{4} - 12 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.96139	−	0.391068i	0.980697	−	0.195534i
							\(3\)	−0.164573	+	2.99548i	−0.0548575	+	0.998494i
\(5\)	−3.61305	+	3.61305i	−0.722609	+	0.722609i								−0.969136	−	0.246527i	\(-0.920711\pi\)
							0.246527	+	0.969136i	\(0.420711\pi\)
\(7\)		−	12.2792i		−	1.75417i	−0.480338	−	0.877083i	\(-0.659486\pi\)
							0.480338	−	0.877083i	\(-0.340514\pi\)
\(11\)	1.76932	−	1.76932i	0.160847	−	0.160847i	−0.622095	−	0.782942i	\(-0.713718\pi\)
							0.782942	+	0.622095i	\(0.213718\pi\)
\(13\)	−2.38826	+	2.38826i	−0.183713	+	0.183713i	−0.792971	−	0.609259i	\(-0.791467\pi\)
							0.609259	+	0.792971i	\(0.291467\pi\)
\(17\)			20.0754i			1.18091i	0.807072	+	0.590453i	\(0.201051\pi\)
							−0.807072	+	0.590453i	\(0.798949\pi\)
\(19\)	−8.77090	+	8.77090i	−0.461626	+	0.461626i	−0.899188	−	0.437562i	\(-0.855842\pi\)
							0.437562	+	0.899188i	\(0.355842\pi\)
\(23\)	−13.1821			−0.573134			−0.286567	−	0.958060i	\(-0.592514\pi\)
							−0.286567	+	0.958060i	\(0.592514\pi\)
\(29\)	−6.51544	−	6.51544i	−0.224670	−	0.224670i	0.585792	−	0.810462i	\(-0.300784\pi\)
							−0.810462	+	0.585792i	\(0.800784\pi\)
\(31\)	37.5922			1.21265			0.606326	−	0.795216i	\(-0.292643\pi\)
							0.606326	+	0.795216i	\(0.292643\pi\)
\(37\)	10.0057	+	10.0057i	0.270423	+	0.270423i	0.829271	−	0.558847i	\(-0.188756\pi\)
							−0.558847	+	0.829271i	\(0.688756\pi\)
\(41\)	−4.57407			−0.111563			−0.0557814	−	0.998443i	\(-0.517765\pi\)
							−0.0557814	+	0.998443i	\(0.517765\pi\)
\(43\)	21.2835	+	21.2835i	0.494966	+	0.494966i	0.909867	−	0.414901i	\(-0.136184\pi\)
							−0.414901	+	0.909867i	\(0.636184\pi\)
\(47\)		−	54.8366i		−	1.16674i	−0.812208	−	0.583368i	\(-0.801734\pi\)
							0.812208	−	0.583368i	\(-0.198266\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.3.i.b.5.10	yes	20
3.2	odd	2	inner	48.3.i.b.5.1	✓	20
4.3	odd	2		192.3.i.b.113.5		20
8.3	odd	2		384.3.i.c.353.6		20
8.5	even	2		384.3.i.d.353.5		20
12.11	even	2		192.3.i.b.113.10		20
16.3	odd	4		192.3.i.b.17.10		20
16.5	even	4		384.3.i.d.161.10		20
16.11	odd	4		384.3.i.c.161.1		20
16.13	even	4	inner	48.3.i.b.29.1	yes	20
24.5	odd	2		384.3.i.d.353.10		20
24.11	even	2		384.3.i.c.353.1		20
48.5	odd	4		384.3.i.d.161.5		20
48.11	even	4		384.3.i.c.161.6		20
48.29	odd	4	inner	48.3.i.b.29.10	yes	20
48.35	even	4		192.3.i.b.17.5		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.3.i.b.5.1	✓	20	3.2	odd	2	inner
48.3.i.b.5.10	yes	20	1.1	even	1	trivial
48.3.i.b.29.1	yes	20	16.13	even	4	inner
48.3.i.b.29.10	yes	20	48.29	odd	4	inner
192.3.i.b.17.5		20	48.35	even	4
192.3.i.b.17.10		20	16.3	odd	4
192.3.i.b.113.5		20	4.3	odd	2
192.3.i.b.113.10		20	12.11	even	2
384.3.i.c.161.1		20	16.11	odd	4
384.3.i.c.161.6		20	48.11	even	4
384.3.i.c.353.1		20	24.11	even	2
384.3.i.c.353.6		20	8.3	odd	2
384.3.i.d.161.5		20	48.5	odd	4
384.3.i.d.161.10		20	16.5	even	4
384.3.i.d.353.5		20	8.5	even	2
384.3.i.d.353.10		20	24.5	odd	2