Embedded newform 56.3.j.a.5.7

• Label: 56.3.j.a.5.7
• 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			5.7
Character	\(\chi\)	\(=\)	56.5
Dual form			56.3.j.a.45.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.212190 - 1.98871i) q^{2} +(-1.16781 - 2.02271i) q^{3} +(-3.90995 + 0.843971i) q^{4} +(1.55055 - 2.68563i) q^{5} +(-3.77480 + 2.75165i) q^{6} +(-6.89374 - 1.21502i) q^{7} +(2.50807 + 7.59668i) q^{8} +(1.77242 - 3.06992i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 2 q^{2} - 4 q^{4} - 4 q^{7} - 20 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.212190	−	1.98871i	−0.106095	−	0.994356i
							\(3\)	−1.16781	−	2.02271i	−0.389271	−	0.674238i	0.603080	−	0.797680i	\(-0.293940\pi\)
−0.992352	+	0.123443i	\(0.960607\pi\)
\(5\)	1.55055	−	2.68563i	0.310110	−	0.537126i	−0.668276	−	0.743913i	\(-0.732968\pi\)
							0.978386	+	0.206787i	\(0.0663009\pi\)
\(7\)	−6.89374	−	1.21502i	−0.984821	−	0.173575i
							\(11\)	4.06604	−	2.34753i	0.369640	−	0.213412i	−0.303661	−	0.952780i	\(-0.598209\pi\)
0.673301	+	0.739368i	\(0.264876\pi\)
\(13\)	6.88097			0.529305			0.264653	−	0.964344i	\(-0.414743\pi\)
							0.264653	+	0.964344i	\(0.414743\pi\)
\(17\)	14.7184	−	8.49765i	0.865786	−	0.499862i	−0.000159428	−	1.00000i	\(-0.500051\pi\)
							0.865946	+	0.500138i	\(0.166717\pi\)
\(19\)	13.1099	−	22.7070i	0.689994	−	1.19510i	−0.281845	−	0.959460i	\(-0.590947\pi\)
							0.971839	−	0.235645i	\(-0.0757201\pi\)
\(23\)	−12.9403	+	22.4132i	−0.562620	+	0.974486i	0.434647	+	0.900601i	\(0.356873\pi\)
							−0.997267	+	0.0738851i	\(0.976460\pi\)
\(29\)			42.2701i			1.45759i	0.684732	+	0.728795i	\(0.259919\pi\)
							−0.684732	+	0.728795i	\(0.740081\pi\)
\(31\)	15.9024	−	9.18126i	0.512981	−	0.296170i	−0.221077	−	0.975256i	\(-0.570957\pi\)
							0.734058	+	0.679087i	\(0.237624\pi\)
\(37\)	−43.1997	−	24.9413i	−1.16756	−	0.674090i	−0.214455	−	0.976734i	\(-0.568797\pi\)
							−0.953104	+	0.302644i	\(0.902131\pi\)
\(41\)		−	10.7844i		−	0.263035i	−0.991314	−	0.131517i	\(-0.958015\pi\)
							0.991314	−	0.131517i	\(-0.0419849\pi\)
\(43\)			24.1791i			0.562304i	0.959663	+	0.281152i	\(0.0907165\pi\)
							−0.959663	+	0.281152i	\(0.909284\pi\)
\(47\)	11.8480	+	6.84046i	0.252086	+	0.145542i	0.620719	−	0.784033i	\(-0.286841\pi\)
							−0.368633	+	0.929575i	\(0.620174\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	56.3.j.a.5.7	✓	28
4.3	odd	2		224.3.n.a.145.10		28
7.2	even	3		392.3.h.a.293.8		28
7.3	odd	6	inner	56.3.j.a.45.13	yes	28
7.4	even	3		392.3.j.e.325.13		28
7.5	odd	6		392.3.h.a.293.7		28
7.6	odd	2		392.3.j.e.117.7		28
8.3	odd	2		224.3.n.a.145.5		28
8.5	even	2	inner	56.3.j.a.5.13	yes	28
28.3	even	6		224.3.n.a.17.5		28
28.19	even	6		1568.3.h.a.881.20		28
28.23	odd	6		1568.3.h.a.881.10		28
56.3	even	6		224.3.n.a.17.10		28
56.5	odd	6		392.3.h.a.293.6		28
56.13	odd	2		392.3.j.e.117.13		28
56.19	even	6		1568.3.h.a.881.9		28
56.37	even	6		392.3.h.a.293.5		28
56.45	odd	6	inner	56.3.j.a.45.7	yes	28
56.51	odd	6		1568.3.h.a.881.19		28
56.53	even	6		392.3.j.e.325.7		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.j.a.5.7	✓	28	1.1	even	1	trivial
56.3.j.a.5.13	yes	28	8.5	even	2	inner
56.3.j.a.45.7	yes	28	56.45	odd	6	inner
56.3.j.a.45.13	yes	28	7.3	odd	6	inner
224.3.n.a.17.5		28	28.3	even	6
224.3.n.a.17.10		28	56.3	even	6
224.3.n.a.145.5		28	8.3	odd	2
224.3.n.a.145.10		28	4.3	odd	2
392.3.h.a.293.5		28	56.37	even	6
392.3.h.a.293.6		28	56.5	odd	6
392.3.h.a.293.7		28	7.5	odd	6
392.3.h.a.293.8		28	7.2	even	3
392.3.j.e.117.7		28	7.6	odd	2
392.3.j.e.117.13		28	56.13	odd	2
392.3.j.e.325.7		28	56.53	even	6
392.3.j.e.325.13		28	7.4	even	3
1568.3.h.a.881.9		28	56.19	even	6
1568.3.h.a.881.10		28	28.23	odd	6
1568.3.h.a.881.19		28	56.51	odd	6
1568.3.h.a.881.20		28	28.19	even	6