# Properties

 Label 2592.2.i.i Level $2592$ Weight $2$ Character orbit 2592.i Analytic conductor $20.697$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 864) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{5} + ( -3 + 3 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q -\zeta_{6} q^{5} + ( -3 + 3 \zeta_{6} ) q^{7} + ( -3 + 3 \zeta_{6} ) q^{11} -4 q^{17} + 6 q^{19} + 6 \zeta_{6} q^{23} + ( 4 - 4 \zeta_{6} ) q^{25} + ( -2 + 2 \zeta_{6} ) q^{29} -9 \zeta_{6} q^{31} + 3 q^{35} -2 q^{37} -10 \zeta_{6} q^{41} + ( -6 + 6 \zeta_{6} ) q^{43} + ( 6 - 6 \zeta_{6} ) q^{47} -2 \zeta_{6} q^{49} -13 q^{53} + 3 q^{55} -12 \zeta_{6} q^{59} + ( -8 + 8 \zeta_{6} ) q^{61} -6 \zeta_{6} q^{67} -12 q^{71} + 9 q^{73} -9 \zeta_{6} q^{77} + ( -3 + 3 \zeta_{6} ) q^{83} + 4 \zeta_{6} q^{85} -14 q^{89} -6 \zeta_{6} q^{95} + ( 9 - 9 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{5} - 3 q^{7} + O(q^{10})$$ $$2 q - q^{5} - 3 q^{7} - 3 q^{11} - 8 q^{17} + 12 q^{19} + 6 q^{23} + 4 q^{25} - 2 q^{29} - 9 q^{31} + 6 q^{35} - 4 q^{37} - 10 q^{41} - 6 q^{43} + 6 q^{47} - 2 q^{49} - 26 q^{53} + 6 q^{55} - 12 q^{59} - 8 q^{61} - 6 q^{67} - 24 q^{71} + 18 q^{73} - 9 q^{77} - 3 q^{83} + 4 q^{85} - 28 q^{89} - 6 q^{95} + 9 q^{97} + O(q^{100})$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 −0.500000 0.866025i 0 −1.50000 + 2.59808i 0 0 0
1729.1 0 0 0 −0.500000 + 0.866025i 0 −1.50000 2.59808i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2592.2.i.i 2
3.b odd 2 1 2592.2.i.m 2
4.b odd 2 1 2592.2.i.l 2
9.c even 3 1 864.2.a.h yes 1
9.c even 3 1 inner 2592.2.i.i 2
9.d odd 6 1 864.2.a.f yes 1
9.d odd 6 1 2592.2.i.m 2
12.b even 2 1 2592.2.i.p 2
36.f odd 6 1 864.2.a.g yes 1
36.f odd 6 1 2592.2.i.l 2
36.h even 6 1 864.2.a.e 1
36.h even 6 1 2592.2.i.p 2
72.j odd 6 1 1728.2.a.t 1
72.l even 6 1 1728.2.a.q 1
72.n even 6 1 1728.2.a.k 1
72.p odd 6 1 1728.2.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.a.e 1 36.h even 6 1
864.2.a.f yes 1 9.d odd 6 1
864.2.a.g yes 1 36.f odd 6 1
864.2.a.h yes 1 9.c even 3 1
1728.2.a.j 1 72.p odd 6 1
1728.2.a.k 1 72.n even 6 1
1728.2.a.q 1 72.l even 6 1
1728.2.a.t 1 72.j odd 6 1
2592.2.i.i 2 1.a even 1 1 trivial
2592.2.i.i 2 9.c even 3 1 inner
2592.2.i.l 2 4.b odd 2 1
2592.2.i.l 2 36.f odd 6 1
2592.2.i.m 2 3.b odd 2 1
2592.2.i.m 2 9.d odd 6 1
2592.2.i.p 2 12.b even 2 1
2592.2.i.p 2 36.h even 6 1

## Hecke kernels

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

 $$T_{5}^{2} + T_{5} + 1$$ $$T_{7}^{2} + 3 T_{7} + 9$$ $$T_{11}^{2} + 3 T_{11} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$9 + 3 T + T^{2}$$
$11$ $$9 + 3 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$( 4 + T )^{2}$$
$19$ $$( -6 + T )^{2}$$
$23$ $$36 - 6 T + T^{2}$$
$29$ $$4 + 2 T + T^{2}$$
$31$ $$81 + 9 T + T^{2}$$
$37$ $$( 2 + T )^{2}$$
$41$ $$100 + 10 T + T^{2}$$
$43$ $$36 + 6 T + T^{2}$$
$47$ $$36 - 6 T + T^{2}$$
$53$ $$( 13 + T )^{2}$$
$59$ $$144 + 12 T + T^{2}$$
$61$ $$64 + 8 T + T^{2}$$
$67$ $$36 + 6 T + T^{2}$$
$71$ $$( 12 + T )^{2}$$
$73$ $$( -9 + T )^{2}$$
$79$ $$T^{2}$$
$83$ $$9 + 3 T + T^{2}$$
$89$ $$( 14 + T )^{2}$$
$97$ $$81 - 9 T + T^{2}$$