Embedded newform 56.3.j.a.5.3

• Label: 56.3.j.a.5.3
• Level: $56$
• Weight: $3$
• Character: 56.5
• Analytic conductor: $1.526$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.52588948042\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(14\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			5.3
Character	\(\chi\)	\(=\)	56.5
Dual form			56.3.j.a.45.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.70738 + 1.04157i) q^{2} +(2.78005 + 4.81519i) q^{3} +(1.83027 - 3.55670i) q^{4} +(-1.52921 + 2.64866i) q^{5} +(-9.76195 - 5.32573i) q^{6} +(0.608243 - 6.97352i) q^{7} +(0.579586 + 7.97898i) q^{8} +(-10.9574 + 18.9787i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 2 q^{2} - 4 q^{4} - 4 q^{7} - 20 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(17\)	\(29\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.70738	+	1.04157i	−0.853688	+	0.520784i
							\(3\)	2.78005	+	4.81519i	0.926684	+	1.60506i	0.788830	+	0.614611i	\(0.210687\pi\)
0.137854	+	0.990453i	\(0.455980\pi\)
\(5\)	−1.52921	+	2.64866i	−0.305841	+	0.529732i	−0.977448	−	0.211175i	\(-0.932271\pi\)
							0.671607	+	0.740907i	\(0.265604\pi\)
\(7\)	0.608243	−	6.97352i	0.0868918	−	0.996218i
							\(11\)	0.106038	−	0.0612210i	0.00963981	−	0.00556554i	−0.495172	−	0.868795i	\(-0.664895\pi\)
0.504812	+	0.863229i	\(0.331562\pi\)
\(13\)	4.11412			0.316471			0.158235	−	0.987401i	\(-0.449420\pi\)
							0.158235	+	0.987401i	\(0.449420\pi\)
\(17\)	17.8551	−	10.3087i	1.05030	−	0.606392i	0.127568	−	0.991830i	\(-0.459283\pi\)
							0.922734	+	0.385438i	\(0.125950\pi\)
\(19\)	4.46893	−	7.74042i	0.235207	−	0.407390i	−0.724126	−	0.689668i	\(-0.757757\pi\)
							0.959333	+	0.282277i	\(0.0910899\pi\)
\(23\)	7.51940	−	13.0240i	0.326930	−	0.566260i	−0.654971	−	0.755654i	\(-0.727319\pi\)
							0.981901	+	0.189394i	\(0.0606524\pi\)
\(29\)			31.6239i			1.09048i	0.838280	+	0.545239i	\(0.183561\pi\)
							−0.838280	+	0.545239i	\(0.816439\pi\)
\(31\)	23.0318	−	13.2974i	0.742962	−	0.428949i	−0.0801833	−	0.996780i	\(-0.525551\pi\)
							0.823145	+	0.567831i	\(0.192217\pi\)
\(37\)	−25.1405	−	14.5149i	−0.679474	−	0.392295i	0.120183	−	0.992752i	\(-0.461652\pi\)
							−0.799657	+	0.600457i	\(0.794985\pi\)
\(41\)		−	9.26915i		−	0.226077i	−0.993591	−	0.113038i	\(-0.963942\pi\)
							0.993591	−	0.113038i	\(-0.0360583\pi\)
\(43\)		−	45.3391i		−	1.05440i	−0.849742	−	0.527199i	\(-0.823242\pi\)
							0.849742	−	0.527199i	\(-0.176758\pi\)
\(47\)	−68.6931	−	39.6600i	−1.46156	−	0.843830i	−0.462472	−	0.886634i	\(-0.653038\pi\)
							−0.999083	+	0.0428039i	\(0.986371\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	56.3.j.a.5.3	✓	28
4.3	odd	2		224.3.n.a.145.1		28
7.2	even	3		392.3.h.a.293.23		28
7.3	odd	6	inner	56.3.j.a.45.8	yes	28
7.4	even	3		392.3.j.e.325.8		28
7.5	odd	6		392.3.h.a.293.24		28
7.6	odd	2		392.3.j.e.117.3		28
8.3	odd	2		224.3.n.a.145.14		28
8.5	even	2	inner	56.3.j.a.5.8	yes	28
28.3	even	6		224.3.n.a.17.14		28
28.19	even	6		1568.3.h.a.881.2		28
28.23	odd	6		1568.3.h.a.881.28		28
56.3	even	6		224.3.n.a.17.1		28
56.5	odd	6		392.3.h.a.293.21		28
56.13	odd	2		392.3.j.e.117.8		28
56.19	even	6		1568.3.h.a.881.27		28
56.37	even	6		392.3.h.a.293.22		28
56.45	odd	6	inner	56.3.j.a.45.3	yes	28
56.51	odd	6		1568.3.h.a.881.1		28
56.53	even	6		392.3.j.e.325.3		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.j.a.5.3	✓	28	1.1	even	1	trivial
56.3.j.a.5.8	yes	28	8.5	even	2	inner
56.3.j.a.45.3	yes	28	56.45	odd	6	inner
56.3.j.a.45.8	yes	28	7.3	odd	6	inner
224.3.n.a.17.1		28	56.3	even	6
224.3.n.a.17.14		28	28.3	even	6
224.3.n.a.145.1		28	4.3	odd	2
224.3.n.a.145.14		28	8.3	odd	2
392.3.h.a.293.21		28	56.5	odd	6
392.3.h.a.293.22		28	56.37	even	6
392.3.h.a.293.23		28	7.2	even	3
392.3.h.a.293.24		28	7.5	odd	6
392.3.j.e.117.3		28	7.6	odd	2
392.3.j.e.117.8		28	56.13	odd	2
392.3.j.e.325.3		28	56.53	even	6
392.3.j.e.325.8		28	7.4	even	3
1568.3.h.a.881.1		28	56.51	odd	6
1568.3.h.a.881.2		28	28.19	even	6
1568.3.h.a.881.27		28	56.19	even	6
1568.3.h.a.881.28		28	28.23	odd	6