Embedded newform 2592.2.i.e.865.1

• Label: 2592.2.i.e.865.1
• Level: $2592$
• Weight: $2$
• Character: 2592.865
• Analytic conductor: $20.697$
• Analytic rank: $1$
• Dimension: $2$
• CM discriminant: -4
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2592 = 2^{5} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2592.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(20.6972242039\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 32)
Sato-Tate group:	$\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label			865.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	2592.865
Dual form			2592.2.i.e.1729.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.73205i) q^{5} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(1217\)	\(2431\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(1\)

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			0
							\(3\)		0			0
\(5\)	−1.00000	−	1.73205i	−0.447214	−	0.774597i								0.550990	−	0.834512i	\(-0.314250\pi\)
							−0.998203	+	0.0599153i	\(0.980917\pi\)
\(7\)		0			0		−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)
\(11\)		0			0		−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)
\(13\)	−3.00000	−	5.19615i	−0.832050	−	1.44115i	−0.896410	−	0.443227i	\(-0.853834\pi\)
							0.0643593	−	0.997927i	\(-0.479500\pi\)
\(17\)	−2.00000			−0.485071			−0.242536	−	0.970143i	\(-0.577979\pi\)
							−0.242536	+	0.970143i	\(0.577979\pi\)
\(19\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(23\)		0			0		0.866025	−	0.500000i	\(-0.166667\pi\)
							−0.866025	+	0.500000i	\(0.833333\pi\)
\(29\)	−5.00000	+	8.66025i	−0.928477	+	1.60817i	−0.142605	+	0.989780i	\(0.545548\pi\)
							−0.785872	+	0.618389i	\(0.787786\pi\)
\(31\)		0			0		0.866025	−	0.500000i	\(-0.166667\pi\)
							−0.866025	+	0.500000i	\(0.833333\pi\)
\(37\)	−2.00000			−0.328798			−0.164399	−	0.986394i	\(-0.552568\pi\)
							−0.164399	+	0.986394i	\(0.552568\pi\)
\(41\)	5.00000	+	8.66025i	0.780869	+	1.35250i	0.931436	+	0.363905i	\(0.118557\pi\)
							−0.150567	+	0.988600i	\(0.548110\pi\)
\(43\)		0			0		−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)
\(47\)		0			0		−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2592.2.i.e.865.1		2
3.2	odd	2		2592.2.i.t.865.1		2
4.3	odd	2	CM	2592.2.i.e.865.1		2
9.2	odd	6		32.2.a.a.1.1	✓	1
9.4	even	3	inner	2592.2.i.e.1729.1		2
9.5	odd	6		2592.2.i.t.1729.1		2
9.7	even	3		288.2.a.d.1.1		1
12.11	even	2		2592.2.i.t.865.1		2
36.7	odd	6		288.2.a.d.1.1		1
36.11	even	6		32.2.a.a.1.1	✓	1
36.23	even	6		2592.2.i.t.1729.1		2
36.31	odd	6	inner	2592.2.i.e.1729.1		2
45.2	even	12		800.2.c.e.449.1		2
45.7	odd	12		7200.2.f.m.6049.1		2
45.29	odd	6		800.2.a.d.1.1		1
45.34	even	6		7200.2.a.v.1.1		1
45.38	even	12		800.2.c.e.449.2		2
45.43	odd	12		7200.2.f.m.6049.2		2
63.2	odd	6		1568.2.i.g.1537.1		2
63.11	odd	6		1568.2.i.g.961.1		2
63.20	even	6		1568.2.a.e.1.1		1
63.38	even	6		1568.2.i.f.961.1		2
63.47	even	6		1568.2.i.f.1537.1		2
72.11	even	6		64.2.a.a.1.1		1
72.29	odd	6		64.2.a.a.1.1		1
72.43	odd	6		576.2.a.c.1.1		1
72.61	even	6		576.2.a.c.1.1		1
99.65	even	6		3872.2.a.f.1.1		1
117.38	odd	6		5408.2.a.g.1.1		1
144.11	even	12		256.2.b.b.129.1		2
144.29	odd	12		256.2.b.b.129.2		2
144.43	odd	12		2304.2.d.j.1153.2		2
144.61	even	12		2304.2.d.j.1153.1		2
144.83	even	12		256.2.b.b.129.2		2
144.101	odd	12		256.2.b.b.129.1		2
144.115	odd	12		2304.2.d.j.1153.1		2
144.133	even	12		2304.2.d.j.1153.2		2
153.101	odd	6		9248.2.a.f.1.1		1
180.7	even	12		7200.2.f.m.6049.1		2
180.43	even	12		7200.2.f.m.6049.2		2
180.47	odd	12		800.2.c.e.449.1		2
180.79	odd	6		7200.2.a.v.1.1		1
180.83	odd	12		800.2.c.e.449.2		2
180.119	even	6		800.2.a.d.1.1		1
252.11	even	6		1568.2.i.g.961.1		2
252.47	odd	6		1568.2.i.f.1537.1		2
252.83	odd	6		1568.2.a.e.1.1		1
252.191	even	6		1568.2.i.g.1537.1		2
252.227	odd	6		1568.2.i.f.961.1		2
288.11	even	24		1024.2.e.j.769.2		4
288.29	odd	24		1024.2.e.j.257.2		4
288.83	even	24		1024.2.e.j.257.1		4
288.101	odd	24		1024.2.e.j.769.1		4
288.155	even	24		1024.2.e.j.769.1		4
288.173	odd	24		1024.2.e.j.257.1		4
288.227	even	24		1024.2.e.j.257.2		4
288.245	odd	24		1024.2.e.j.769.2		4
360.29	odd	6		1600.2.a.n.1.1		1
360.83	odd	12		1600.2.c.l.449.1		2
360.173	even	12		1600.2.c.l.449.1		2
360.227	odd	12		1600.2.c.l.449.2		2
360.299	even	6		1600.2.a.n.1.1		1
360.317	even	12		1600.2.c.l.449.2		2
396.263	odd	6		3872.2.a.f.1.1		1
468.155	even	6		5408.2.a.g.1.1		1
504.83	odd	6		3136.2.a.m.1.1		1
504.461	even	6		3136.2.a.m.1.1		1
612.407	even	6		9248.2.a.f.1.1		1
792.461	even	6		7744.2.a.v.1.1		1
792.659	odd	6		7744.2.a.v.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.2.a.a.1.1	✓	1	9.2	odd	6
32.2.a.a.1.1	✓	1	36.11	even	6
64.2.a.a.1.1		1	72.11	even	6
64.2.a.a.1.1		1	72.29	odd	6
256.2.b.b.129.1		2	144.11	even	12
256.2.b.b.129.1		2	144.101	odd	12
256.2.b.b.129.2		2	144.29	odd	12
256.2.b.b.129.2		2	144.83	even	12
288.2.a.d.1.1		1	9.7	even	3
288.2.a.d.1.1		1	36.7	odd	6
576.2.a.c.1.1		1	72.43	odd	6
576.2.a.c.1.1		1	72.61	even	6
800.2.a.d.1.1		1	45.29	odd	6
800.2.a.d.1.1		1	180.119	even	6
800.2.c.e.449.1		2	45.2	even	12
800.2.c.e.449.1		2	180.47	odd	12
800.2.c.e.449.2		2	45.38	even	12
800.2.c.e.449.2		2	180.83	odd	12
1024.2.e.j.257.1		4	288.83	even	24
1024.2.e.j.257.1		4	288.173	odd	24
1024.2.e.j.257.2		4	288.29	odd	24
1024.2.e.j.257.2		4	288.227	even	24
1024.2.e.j.769.1		4	288.101	odd	24
1024.2.e.j.769.1		4	288.155	even	24
1024.2.e.j.769.2		4	288.11	even	24
1024.2.e.j.769.2		4	288.245	odd	24
1568.2.a.e.1.1		1	63.20	even	6
1568.2.a.e.1.1		1	252.83	odd	6
1568.2.i.f.961.1		2	63.38	even	6
1568.2.i.f.961.1		2	252.227	odd	6
1568.2.i.f.1537.1		2	63.47	even	6
1568.2.i.f.1537.1		2	252.47	odd	6
1568.2.i.g.961.1		2	63.11	odd	6
1568.2.i.g.961.1		2	252.11	even	6
1568.2.i.g.1537.1		2	63.2	odd	6
1568.2.i.g.1537.1		2	252.191	even	6
1600.2.a.n.1.1		1	360.29	odd	6
1600.2.a.n.1.1		1	360.299	even	6
1600.2.c.l.449.1		2	360.83	odd	12
1600.2.c.l.449.1		2	360.173	even	12
1600.2.c.l.449.2		2	360.227	odd	12
1600.2.c.l.449.2		2	360.317	even	12
2304.2.d.j.1153.1		2	144.61	even	12
2304.2.d.j.1153.1		2	144.115	odd	12
2304.2.d.j.1153.2		2	144.43	odd	12
2304.2.d.j.1153.2		2	144.133	even	12
2592.2.i.e.865.1		2	1.1	even	1	trivial
2592.2.i.e.865.1		2	4.3	odd	2	CM
2592.2.i.e.1729.1		2	9.4	even	3	inner
2592.2.i.e.1729.1		2	36.31	odd	6	inner
2592.2.i.t.865.1		2	3.2	odd	2
2592.2.i.t.865.1		2	12.11	even	2
2592.2.i.t.1729.1		2	9.5	odd	6
2592.2.i.t.1729.1		2	36.23	even	6
3136.2.a.m.1.1		1	504.83	odd	6
3136.2.a.m.1.1		1	504.461	even	6
3872.2.a.f.1.1		1	99.65	even	6
3872.2.a.f.1.1		1	396.263	odd	6
5408.2.a.g.1.1		1	117.38	odd	6
5408.2.a.g.1.1		1	468.155	even	6
7200.2.a.v.1.1		1	45.34	even	6
7200.2.a.v.1.1		1	180.79	odd	6
7200.2.f.m.6049.1		2	45.7	odd	12
7200.2.f.m.6049.1		2	180.7	even	12
7200.2.f.m.6049.2		2	45.43	odd	12
7200.2.f.m.6049.2		2	180.43	even	12
7744.2.a.v.1.1		1	792.461	even	6
7744.2.a.v.1.1		1	792.659	odd	6
9248.2.a.f.1.1		1	153.101	odd	6
9248.2.a.f.1.1		1	612.407	even	6