# Properties

 Label 2592.2.a.x Level $2592$ Weight $2$ Character orbit 2592.a Self dual yes Analytic conductor $20.697$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2592 = 2^{5} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2592.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$20.6972242039$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.13068.1 Defining polynomial: $$x^{4} - x^{3} - 6 x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: no (minimal twist has level 288) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{1} ) q^{5} + \beta_{3} q^{7} +O(q^{10})$$ $$q + ( 1 + \beta_{1} ) q^{5} + \beta_{3} q^{7} + \beta_{2} q^{11} + ( 1 - \beta_{1} ) q^{13} + \beta_{1} q^{17} + ( -\beta_{2} + \beta_{3} ) q^{19} -\beta_{3} q^{23} + ( 4 + \beta_{1} ) q^{25} + ( 3 + \beta_{1} ) q^{29} + ( 2 \beta_{2} - \beta_{3} ) q^{31} + ( -2 \beta_{2} + 3 \beta_{3} ) q^{35} + 4 q^{37} - q^{41} + \beta_{2} q^{43} -\beta_{3} q^{47} + ( 8 + 3 \beta_{1} ) q^{49} + 4 q^{53} + ( -2 \beta_{2} - \beta_{3} ) q^{55} + ( -\beta_{2} - 2 \beta_{3} ) q^{59} + ( 5 - 3 \beta_{1} ) q^{61} + ( -7 + \beta_{1} ) q^{65} + ( -\beta_{2} - 2 \beta_{3} ) q^{67} -2 \beta_{3} q^{71} + ( 8 + \beta_{1} ) q^{73} + ( 3 - 3 \beta_{1} ) q^{77} + ( -2 \beta_{2} + \beta_{3} ) q^{79} + ( 2 \beta_{2} + \beta_{3} ) q^{83} + 8 q^{85} + ( -8 - 2 \beta_{1} ) q^{89} + ( 2 \beta_{2} - \beta_{3} ) q^{91} + 4 \beta_{3} q^{95} + 9 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{5} + O(q^{10})$$ $$4q + 2q^{5} + 6q^{13} - 2q^{17} + 14q^{25} + 10q^{29} + 16q^{37} - 4q^{41} + 26q^{49} + 16q^{53} + 26q^{61} - 30q^{65} + 30q^{73} + 18q^{77} + 32q^{85} - 28q^{89} + 36q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 6 x^{2} - x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} - \nu^{2} - 7 \nu - 1$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - 6 \nu - 4$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{3} - 3 \nu^{2} - 9 \nu + 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - 3 \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + 5 \beta_{2} - 3 \beta_{1} + 18$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2 \beta_{2} - 3 \beta_{1} + 4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0.328543 3.04374 −1.82405 −0.548230
0 0 0 −2.37228 0 −2.20979 0 0 0
1.2 0 0 0 −2.37228 0 2.20979 0 0 0
1.3 0 0 0 3.37228 0 −4.70285 0 0 0
1.4 0 0 0 3.37228 0 4.70285 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2592.2.a.x 4
3.b odd 2 1 2592.2.a.u 4
4.b odd 2 1 inner 2592.2.a.x 4
8.b even 2 1 5184.2.a.cc 4
8.d odd 2 1 5184.2.a.cc 4
9.c even 3 2 864.2.i.f 8
9.d odd 6 2 288.2.i.f 8
12.b even 2 1 2592.2.a.u 4
24.f even 2 1 5184.2.a.cf 4
24.h odd 2 1 5184.2.a.cf 4
36.f odd 6 2 864.2.i.f 8
36.h even 6 2 288.2.i.f 8
72.j odd 6 2 576.2.i.n 8
72.l even 6 2 576.2.i.n 8
72.n even 6 2 1728.2.i.n 8
72.p odd 6 2 1728.2.i.n 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.f 8 9.d odd 6 2
288.2.i.f 8 36.h even 6 2
576.2.i.n 8 72.j odd 6 2
576.2.i.n 8 72.l even 6 2
864.2.i.f 8 9.c even 3 2
864.2.i.f 8 36.f odd 6 2
1728.2.i.n 8 72.n even 6 2
1728.2.i.n 8 72.p odd 6 2
2592.2.a.u 4 3.b odd 2 1
2592.2.a.u 4 12.b even 2 1
2592.2.a.x 4 1.a even 1 1 trivial
2592.2.a.x 4 4.b odd 2 1 inner
5184.2.a.cc 4 8.b even 2 1
5184.2.a.cc 4 8.d odd 2 1
5184.2.a.cf 4 24.f even 2 1
5184.2.a.cf 4 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2592))$$:

 $$T_{5}^{2} - T_{5} - 8$$ $$T_{7}^{4} - 27 T_{7}^{2} + 108$$ $$T_{11}^{4} - 36 T_{11}^{2} + 27$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -8 - T + T^{2} )^{2}$$
$7$ $$108 - 27 T^{2} + T^{4}$$
$11$ $$27 - 36 T^{2} + T^{4}$$
$13$ $$( -6 - 3 T + T^{2} )^{2}$$
$17$ $$( -8 + T + T^{2} )^{2}$$
$19$ $$432 - 45 T^{2} + T^{4}$$
$23$ $$108 - 27 T^{2} + T^{4}$$
$29$ $$( -2 - 5 T + T^{2} )^{2}$$
$31$ $$3888 - 135 T^{2} + T^{4}$$
$37$ $$( -4 + T )^{4}$$
$41$ $$( 1 + T )^{4}$$
$43$ $$27 - 36 T^{2} + T^{4}$$
$47$ $$108 - 27 T^{2} + T^{4}$$
$53$ $$( -4 + T )^{4}$$
$59$ $$7803 - 180 T^{2} + T^{4}$$
$61$ $$( -32 - 13 T + T^{2} )^{2}$$
$67$ $$7803 - 180 T^{2} + T^{4}$$
$71$ $$1728 - 108 T^{2} + T^{4}$$
$73$ $$( 48 - 15 T + T^{2} )^{2}$$
$79$ $$3888 - 135 T^{2} + T^{4}$$
$83$ $$1728 - 207 T^{2} + T^{4}$$
$89$ $$( 16 + 14 T + T^{2} )^{2}$$
$97$ $$( -9 + T )^{4}$$