Embedded newform 56.5.k.a.11.15

• Label: 56.5.k.a.11.15
• Level: $56$
• Weight: $5$
• Character: 56.11
• Analytic conductor: $5.789$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 56 = 2^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	56.k (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.78871793270\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(30\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.15
Character	\(\chi\)	\(=\)	56.11
Dual form			56.5.k.a.51.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.229998 + 3.99338i) q^{2} +(7.74459 - 13.4140i) q^{3} +(-15.8942 - 1.83694i) q^{4} +(-13.5276 + 7.81018i) q^{5} +(51.7861 + 34.0123i) q^{6} +(43.5544 - 22.4503i) q^{7} +(10.9912 - 63.0491i) q^{8} +(-79.4574 - 137.624i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 4 q^{2} - 2 q^{3} - 4 q^{4} + 28 q^{6} - 4 q^{8} - 704 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/56\mathbb{Z}\right)^\times\).

\(n\)	\(15\)	\(17\)	\(29\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.229998	+	3.99338i	−0.0574995	+	0.998346i
							\(3\)	7.74459	−	13.4140i	0.860510	−	1.49045i	−0.0109273	−	0.999940i	\(-0.503478\pi\)
0.871437	−	0.490507i	\(-0.163188\pi\)
\(5\)	−13.5276	+	7.81018i	−0.541105	+	0.312407i	−0.745527	−	0.666476i	\(-0.767802\pi\)
							0.204422	+	0.978883i	\(0.434469\pi\)
\(7\)	43.5544	−	22.4503i	0.888865	−	0.458170i
							\(11\)	81.1739	−	140.597i	0.670859	−	1.16196i	−0.306802	−	0.951773i	\(-0.599259\pi\)
0.977661	−	0.210188i	\(-0.0674076\pi\)
\(13\)		−	187.067i		−	1.10691i	−0.832880	−	0.553453i	\(-0.813310\pi\)
							0.832880	−	0.553453i	\(-0.186690\pi\)
\(17\)	−128.409	+	222.411i	−0.444323	+	0.769589i	−0.998005	−	0.0631387i	\(-0.979889\pi\)
							0.553682	+	0.832728i	\(0.313222\pi\)
\(19\)	97.3921	+	168.688i	0.269784	+	0.467280i	0.968806	−	0.247820i	\(-0.0797143\pi\)
							−0.699022	+	0.715100i	\(0.746381\pi\)
\(23\)	−7.27105	+	4.19794i	−0.0137449	+	0.00793562i	−0.506857	−	0.862030i	\(-0.669193\pi\)
							0.493112	+	0.869966i	\(0.335859\pi\)
\(29\)			1453.99i			1.72888i	0.502736	+	0.864440i	\(0.332327\pi\)
							−0.502736	+	0.864440i	\(0.667673\pi\)
\(31\)	−330.503	−	190.816i	−0.343916	−	0.198560i	0.318086	−	0.948062i	\(-0.396960\pi\)
							−0.662002	+	0.749502i	\(0.730293\pi\)
\(37\)	2295.71	−	1325.43i	1.67693	−	0.968173i	0.713322	−	0.700836i	\(-0.247190\pi\)
							0.963603	−	0.267337i	\(-0.0861438\pi\)
\(41\)	434.605			0.258539			0.129270	−	0.991609i	\(-0.458737\pi\)
							0.129270	+	0.991609i	\(0.458737\pi\)
\(43\)	291.635			0.157726			0.0788628	−	0.996885i	\(-0.474871\pi\)
							0.0788628	+	0.996885i	\(0.474871\pi\)
\(47\)	−93.1673	+	53.7902i	−0.0421762	+	0.0243505i	−0.520940	−	0.853593i	\(-0.674418\pi\)
							0.478764	+	0.877944i	\(0.341085\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	56.5.k.a.11.15	✓	60
4.3	odd	2		224.5.o.a.207.1		60
7.2	even	3	inner	56.5.k.a.51.27	yes	60
8.3	odd	2	inner	56.5.k.a.11.27	yes	60
8.5	even	2		224.5.o.a.207.2		60
28.23	odd	6		224.5.o.a.79.2		60
56.37	even	6		224.5.o.a.79.1		60
56.51	odd	6	inner	56.5.k.a.51.15	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.5.k.a.11.15	✓	60	1.1	even	1	trivial
56.5.k.a.11.27	yes	60	8.3	odd	2	inner
56.5.k.a.51.15	yes	60	56.51	odd	6	inner
56.5.k.a.51.27	yes	60	7.2	even	3	inner
224.5.o.a.79.1		60	56.37	even	6
224.5.o.a.79.2		60	28.23	odd	6
224.5.o.a.207.1		60	4.3	odd	2
224.5.o.a.207.2		60	8.5	even	2