Embedded newform 48.6.k.a.11.11

• Label: 48.6.k.a.11.11
• Level: $48$
• Weight: $6$
• Character: 48.11
• Analytic conductor: $7.698$
• Analytic rank: $0$
• Dimension: $76$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	48.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.69842335102\)
Analytic rank:	\(0\)
Dimension:	\(76\)
Relative dimension:	\(38\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			11.11
Character	\(\chi\)	\(=\)	48.11
Dual form			48.6.k.a.35.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.59855 + 4.36468i) q^{2} +(-9.65436 - 12.2390i) q^{3} +(-6.10082 - 31.4131i) q^{4} +(13.2817 - 13.2817i) q^{5} +(88.1611 + 1.90456i) q^{6} -46.0433 q^{7} +(159.062 + 86.4135i) q^{8} +(-56.5866 + 236.320i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 2 q^{3} - 4 q^{4} - 116 q^{6} - 8 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.59855	+	4.36468i	−0.636141	+	0.771573i
							\(3\)	−9.65436	−	12.2390i	−0.619328	−	0.785133i
\(5\)	13.2817	−	13.2817i	0.237590	−	0.237590i								−0.578262	−	0.815851i	\(-0.696269\pi\)
							0.815851	+	0.578262i	\(0.196269\pi\)
\(7\)	−46.0433			−0.355158			−0.177579	−	0.984107i	\(-0.556827\pi\)
							−0.177579	+	0.984107i	\(0.556827\pi\)
\(11\)	53.3105	+	53.3105i	0.132841	+	0.132841i	0.770401	−	0.637560i	\(-0.220056\pi\)
							−0.637560	+	0.770401i	\(0.720056\pi\)
\(13\)	−423.831	+	423.831i	−0.695560	+	0.695560i	−0.963450	−	0.267890i	\(-0.913674\pi\)
							0.267890	+	0.963450i	\(0.413674\pi\)
\(17\)			968.513i			0.812799i	0.913696	+	0.406399i	\(0.133216\pi\)
							−0.913696	+	0.406399i	\(0.866784\pi\)
\(19\)	−710.018	−	710.018i	−0.451217	−	0.451217i	0.444542	−	0.895758i	\(-0.353367\pi\)
							−0.895758	+	0.444542i	\(0.853367\pi\)
\(23\)			3346.97i			1.31927i	0.751588	+	0.659633i	\(0.229288\pi\)
							−0.751588	+	0.659633i	\(0.770712\pi\)
\(29\)	5796.92	+	5796.92i	1.27998	+	1.27998i	0.940678	+	0.339299i	\(0.110190\pi\)
							0.339299	+	0.940678i	\(0.389810\pi\)
\(31\)		−	3760.89i		−	0.702889i	−0.936209	−	0.351445i	\(-0.885691\pi\)
							0.936209	−	0.351445i	\(-0.114309\pi\)
\(37\)	−1252.81	−	1252.81i	−0.150446	−	0.150446i	0.627871	−	0.778317i	\(-0.283926\pi\)
							−0.778317	+	0.627871i	\(0.783926\pi\)
\(41\)	−15306.0			−1.42201			−0.711004	−	0.703188i	\(-0.751759\pi\)
							−0.711004	+	0.703188i	\(0.751759\pi\)
\(43\)	4130.25	−	4130.25i	0.340648	−	0.340648i	−0.515963	−	0.856611i	\(-0.672566\pi\)
							0.856611	+	0.515963i	\(0.172566\pi\)
\(47\)	−12588.7			−0.831260			−0.415630	−	0.909534i	\(-0.636439\pi\)
							−0.415630	+	0.909534i	\(0.636439\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.6.k.a.11.11	✓	76
3.2	odd	2	inner	48.6.k.a.11.28	yes	76
4.3	odd	2		192.6.k.a.143.28		76
12.11	even	2		192.6.k.a.143.31		76
16.3	odd	4	inner	48.6.k.a.35.28	yes	76
16.13	even	4		192.6.k.a.47.31		76
48.29	odd	4		192.6.k.a.47.28		76
48.35	even	4	inner	48.6.k.a.35.11	yes	76

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.6.k.a.11.11	✓	76	1.1	even	1	trivial
48.6.k.a.11.28	yes	76	3.2	odd	2	inner
48.6.k.a.35.11	yes	76	48.35	even	4	inner
48.6.k.a.35.28	yes	76	16.3	odd	4	inner
192.6.k.a.47.28		76	48.29	odd	4
192.6.k.a.47.31		76	16.13	even	4
192.6.k.a.143.28		76	4.3	odd	2
192.6.k.a.143.31		76	12.11	even	2