# Properties

 Label 2592.2.a.j Level $2592$ Weight $2$ Character orbit 2592.a Self dual yes Analytic conductor $20.697$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ Defining polynomial: $$x^{2} - 6$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ 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 $$\beta = \sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{5} + ( -1 + \beta ) q^{7} +O(q^{10})$$ $$q - q^{5} + ( -1 + \beta ) q^{7} + ( 1 + \beta ) q^{11} + ( 1 - 2 \beta ) q^{13} -2 \beta q^{17} -4 q^{19} + ( 3 - \beta ) q^{23} -4 q^{25} + ( 5 + 2 \beta ) q^{29} + ( -5 - \beta ) q^{31} + ( 1 - \beta ) q^{35} + ( 4 + 2 \beta ) q^{37} + ( -7 + 2 \beta ) q^{41} + ( -5 - 3 \beta ) q^{43} + ( -1 - 3 \beta ) q^{47} -2 \beta q^{49} + ( 4 - 2 \beta ) q^{53} + ( -1 - \beta ) q^{55} + ( -7 + 3 \beta ) q^{59} + ( -3 + 2 \beta ) q^{61} + ( -1 + 2 \beta ) q^{65} + ( -5 + 3 \beta ) q^{67} + ( 2 + 4 \beta ) q^{71} + 2 \beta q^{73} + 5 q^{77} + ( -11 + \beta ) q^{79} + ( 3 + \beta ) q^{83} + 2 \beta q^{85} + ( -8 + 2 \beta ) q^{89} + ( -13 + 3 \beta ) q^{91} + 4 q^{95} + ( 1 + 2 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} - 2 q^{7} + O(q^{10})$$ $$2 q - 2 q^{5} - 2 q^{7} + 2 q^{11} + 2 q^{13} - 8 q^{19} + 6 q^{23} - 8 q^{25} + 10 q^{29} - 10 q^{31} + 2 q^{35} + 8 q^{37} - 14 q^{41} - 10 q^{43} - 2 q^{47} + 8 q^{53} - 2 q^{55} - 14 q^{59} - 6 q^{61} - 2 q^{65} - 10 q^{67} + 4 q^{71} + 10 q^{77} - 22 q^{79} + 6 q^{83} - 16 q^{89} - 26 q^{91} + 8 q^{95} + 2 q^{97} + O(q^{100})$$

## 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
 −2.44949 2.44949
0 0 0 −1.00000 0 −3.44949 0 0 0
1.2 0 0 0 −1.00000 0 1.44949 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2592.2.a.j 2
3.b odd 2 1 2592.2.a.o 2
4.b odd 2 1 2592.2.a.n 2
8.b even 2 1 5184.2.a.bu 2
8.d odd 2 1 5184.2.a.by 2
9.c even 3 2 288.2.i.c 4
9.d odd 6 2 864.2.i.e 4
12.b even 2 1 2592.2.a.s 2
24.f even 2 1 5184.2.a.bn 2
24.h odd 2 1 5184.2.a.bj 2
36.f odd 6 2 288.2.i.e yes 4
36.h even 6 2 864.2.i.c 4
72.j odd 6 2 1728.2.i.m 4
72.l even 6 2 1728.2.i.k 4
72.n even 6 2 576.2.i.m 4
72.p odd 6 2 576.2.i.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.c 4 9.c even 3 2
288.2.i.e yes 4 36.f odd 6 2
576.2.i.i 4 72.p odd 6 2
576.2.i.m 4 72.n even 6 2
864.2.i.c 4 36.h even 6 2
864.2.i.e 4 9.d odd 6 2
1728.2.i.k 4 72.l even 6 2
1728.2.i.m 4 72.j odd 6 2
2592.2.a.j 2 1.a even 1 1 trivial
2592.2.a.n 2 4.b odd 2 1
2592.2.a.o 2 3.b odd 2 1
2592.2.a.s 2 12.b even 2 1
5184.2.a.bj 2 24.h odd 2 1
5184.2.a.bn 2 24.f even 2 1
5184.2.a.bu 2 8.b even 2 1
5184.2.a.by 2 8.d odd 2 1

## Hecke kernels

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

 $$T_{5} + 1$$ $$T_{7}^{2} + 2 T_{7} - 5$$ $$T_{11}^{2} - 2 T_{11} - 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$-5 + 2 T + T^{2}$$
$11$ $$-5 - 2 T + T^{2}$$
$13$ $$-23 - 2 T + T^{2}$$
$17$ $$-24 + T^{2}$$
$19$ $$( 4 + T )^{2}$$
$23$ $$3 - 6 T + T^{2}$$
$29$ $$1 - 10 T + T^{2}$$
$31$ $$19 + 10 T + T^{2}$$
$37$ $$-8 - 8 T + T^{2}$$
$41$ $$25 + 14 T + T^{2}$$
$43$ $$-29 + 10 T + T^{2}$$
$47$ $$-53 + 2 T + T^{2}$$
$53$ $$-8 - 8 T + T^{2}$$
$59$ $$-5 + 14 T + T^{2}$$
$61$ $$-15 + 6 T + T^{2}$$
$67$ $$-29 + 10 T + T^{2}$$
$71$ $$-92 - 4 T + T^{2}$$
$73$ $$-24 + T^{2}$$
$79$ $$115 + 22 T + T^{2}$$
$83$ $$3 - 6 T + T^{2}$$
$89$ $$40 + 16 T + T^{2}$$
$97$ $$-23 - 2 T + T^{2}$$