Newform orbit 2592.2.i.e

• Label: 2592.2.i.e
• Level: $2592$
• Weight: $2$
• Character orbit: 2592.i
• 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:	\(\Q(\sqrt{-3}) \)
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:	$\mathrm{U}(1)[D_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - 2 \zeta_{6} q^{5} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5}+O(q^{10}) \)

Character values

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

\(n\)	\(325\)	\(1217\)	\(2431\)
\(\chi(n)\)	\(1\)	\(-\zeta_{6}\)	\(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

865.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

0

0

−1.00000

−

1.73205i

0

0

0

0

0

1729.1

0

0

0

−1.00000

+

1.73205i

0

0

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
9.c	even	3	1	inner
36.f	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2592.2.i.e		2
3.b	odd	2	1		2592.2.i.t		2
4.b	odd	2	1	CM	2592.2.i.e		2
9.c	even	3	1		288.2.a.d		1
9.c	even	3	1	inner	2592.2.i.e		2
9.d	odd	6	1		32.2.a.a	✓	1
9.d	odd	6	1		2592.2.i.t		2
12.b	even	2	1		2592.2.i.t		2
36.f	odd	6	1		288.2.a.d		1
36.f	odd	6	1	inner	2592.2.i.e		2
36.h	even	6	1		32.2.a.a	✓	1
36.h	even	6	1		2592.2.i.t		2
45.h	odd	6	1		800.2.a.d		1
45.j	even	6	1		7200.2.a.v		1
45.k	odd	12	2		7200.2.f.m		2
45.l	even	12	2		800.2.c.e		2
63.i	even	6	1		1568.2.i.f		2
63.j	odd	6	1		1568.2.i.g		2
63.n	odd	6	1		1568.2.i.g		2
63.o	even	6	1		1568.2.a.e		1
63.s	even	6	1		1568.2.i.f		2
72.j	odd	6	1		64.2.a.a		1
72.l	even	6	1		64.2.a.a		1
72.n	even	6	1		576.2.a.c		1
72.p	odd	6	1		576.2.a.c		1
99.g	even	6	1		3872.2.a.f		1
117.n	odd	6	1		5408.2.a.g		1
144.u	even	12	2		256.2.b.b		2
144.v	odd	12	2		2304.2.d.j		2
144.w	odd	12	2		256.2.b.b		2
144.x	even	12	2		2304.2.d.j		2
153.i	odd	6	1		9248.2.a.f		1
180.n	even	6	1		800.2.a.d		1
180.p	odd	6	1		7200.2.a.v		1
180.v	odd	12	2		800.2.c.e		2
180.x	even	12	2		7200.2.f.m		2
252.o	even	6	1		1568.2.i.g		2
252.r	odd	6	1		1568.2.i.f		2
252.s	odd	6	1		1568.2.a.e		1
252.bb	even	6	1		1568.2.i.g		2
252.bn	odd	6	1		1568.2.i.f		2
288.be	odd	24	4		1024.2.e.j		4
288.bf	even	24	4		1024.2.e.j		4
360.bd	even	6	1		1600.2.a.n		1
360.bh	odd	6	1		1600.2.a.n		1
360.br	even	12	2		1600.2.c.l		2
360.bt	odd	12	2		1600.2.c.l		2
396.o	odd	6	1		3872.2.a.f		1
468.x	even	6	1		5408.2.a.g		1
504.cc	even	6	1		3136.2.a.m		1
504.co	odd	6	1		3136.2.a.m		1
612.n	even	6	1		9248.2.a.f		1
792.s	odd	6	1		7744.2.a.v		1
792.w	even	6	1		7744.2.a.v		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
32.2.a.a	✓	1	9.d	odd	6	1
32.2.a.a	✓	1	36.h	even	6	1
64.2.a.a		1	72.j	odd	6	1
64.2.a.a		1	72.l	even	6	1
256.2.b.b		2	144.u	even	12	2
256.2.b.b		2	144.w	odd	12	2
288.2.a.d		1	9.c	even	3	1
288.2.a.d		1	36.f	odd	6	1
576.2.a.c		1	72.n	even	6	1
576.2.a.c		1	72.p	odd	6	1
800.2.a.d		1	45.h	odd	6	1
800.2.a.d		1	180.n	even	6	1
800.2.c.e		2	45.l	even	12	2
800.2.c.e		2	180.v	odd	12	2
1024.2.e.j		4	288.be	odd	24	4
1024.2.e.j		4	288.bf	even	24	4
1568.2.a.e		1	63.o	even	6	1
1568.2.a.e		1	252.s	odd	6	1
1568.2.i.f		2	63.i	even	6	1
1568.2.i.f		2	63.s	even	6	1
1568.2.i.f		2	252.r	odd	6	1
1568.2.i.f		2	252.bn	odd	6	1
1568.2.i.g		2	63.j	odd	6	1
1568.2.i.g		2	63.n	odd	6	1
1568.2.i.g		2	252.o	even	6	1
1568.2.i.g		2	252.bb	even	6	1
1600.2.a.n		1	360.bd	even	6	1
1600.2.a.n		1	360.bh	odd	6	1
1600.2.c.l		2	360.br	even	12	2
1600.2.c.l		2	360.bt	odd	12	2
2304.2.d.j		2	144.v	odd	12	2
2304.2.d.j		2	144.x	even	12	2
2592.2.i.e		2	1.a	even	1	1	trivial
2592.2.i.e		2	4.b	odd	2	1	CM
2592.2.i.e		2	9.c	even	3	1	inner
2592.2.i.e		2	36.f	odd	6	1	inner
2592.2.i.t		2	3.b	odd	2	1
2592.2.i.t		2	9.d	odd	6	1
2592.2.i.t		2	12.b	even	2	1
2592.2.i.t		2	36.h	even	6	1
3136.2.a.m		1	504.cc	even	6	1
3136.2.a.m		1	504.co	odd	6	1
3872.2.a.f		1	99.g	even	6	1
3872.2.a.f		1	396.o	odd	6	1
5408.2.a.g		1	117.n	odd	6	1
5408.2.a.g		1	468.x	even	6	1
7200.2.a.v		1	45.j	even	6	1
7200.2.a.v		1	180.p	odd	6	1
7200.2.f.m		2	45.k	odd	12	2
7200.2.f.m		2	180.x	even	12	2
7744.2.a.v		1	792.s	odd	6	1
7744.2.a.v		1	792.w	even	6	1
9248.2.a.f		1	153.i	odd	6	1
9248.2.a.f		1	612.n	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2592, [\chi])\):

\( T_{5}^{2} + 2T_{5} + 4 \)

\( T_{7} \)

\( T_{11} \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 2T + 4 \)
$7$	\( T^{2} \)
$11$	\( T^{2} \)