Embedded newform 56.5.k.a.51.15

• Label: 56.5.k.a.51.15
• Level: $56$
• Weight: $5$
• Character: 56.51
• 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			51.15
Character	\(\chi\)	\(=\)	56.51
Dual form			56.5.k.a.11.15

$q$-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 \(\left(\mathbb{Z}/56\mathbb{Z}\right)^\times\).

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

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	0.871437	+	0.490507i	\(0.163188\pi\)
−0.0109273	+	0.999940i	\(0.503478\pi\)
\(5\)	−13.5276	−	7.81018i	−0.541105	−	0.312407i	0.204422	−	0.978883i	\(-0.434469\pi\)
							−0.745527	+	0.666476i	\(0.767802\pi\)
\(7\)	43.5544	+	22.4503i	0.888865	+	0.458170i
							\(11\)	81.1739	+	140.597i	0.670859	+	1.16196i	0.977661	+	0.210188i	\(0.0674076\pi\)
−0.306802	+	0.951773i	\(0.599259\pi\)
\(13\)			187.067i			1.10691i	0.832880	+	0.553453i	\(0.186690\pi\)
							−0.832880	+	0.553453i	\(0.813310\pi\)
\(17\)	−128.409	−	222.411i	−0.444323	−	0.769589i	0.553682	−	0.832728i	\(-0.313222\pi\)
							−0.998005	+	0.0631387i	\(0.979889\pi\)
\(19\)	97.3921	−	168.688i	0.269784	−	0.467280i	−0.699022	−	0.715100i	\(-0.746381\pi\)
							0.968806	+	0.247820i	\(0.0797143\pi\)
\(23\)	−7.27105	−	4.19794i	−0.0137449	−	0.00793562i	0.493112	−	0.869966i	\(-0.335859\pi\)
							−0.506857	+	0.862030i	\(0.669193\pi\)
\(29\)		−	1453.99i		−	1.72888i	−0.502736	−	0.864440i	\(-0.667673\pi\)
							0.502736	−	0.864440i	\(-0.332327\pi\)
\(31\)	−330.503	+	190.816i	−0.343916	+	0.198560i	−0.662002	−	0.749502i	\(-0.730293\pi\)
							0.318086	+	0.948062i	\(0.396960\pi\)
\(37\)	2295.71	+	1325.43i	1.67693	+	0.968173i	0.963603	+	0.267337i	\(0.0861438\pi\)
							0.713322	+	0.700836i	\(0.247190\pi\)
\(41\)	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	0.478764	−	0.877944i	\(-0.341085\pi\)
							−0.520940	+	0.853593i	\(0.674418\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.5.k.a.11.15	✓	60	56.11	odd	6	inner
56.5.k.a.11.27	yes	60	7.4	even	3	inner
56.5.k.a.51.15	yes	60	1.1	even	1	trivial
56.5.k.a.51.27	yes	60	8.3	odd	2	inner
224.5.o.a.79.1		60	4.3	odd	2
224.5.o.a.79.2		60	8.5	even	2
224.5.o.a.207.1		60	56.53	even	6
224.5.o.a.207.2		60	28.11	odd	6