# Properties

 Label 2592.2.s.a Level $2592$ Weight $2$ Character orbit 2592.s Analytic conductor $20.697$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2592 = 2^{5} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2592.s (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$20.6972242039$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 864) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 + \zeta_{24} + \zeta_{24}^{4} - \zeta_{24}^{7} ) q^{5} + ( -2 \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{7} +O(q^{10})$$ $$q + ( -2 + \zeta_{24} + \zeta_{24}^{4} - \zeta_{24}^{7} ) q^{5} + ( -2 \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{7} + ( -\zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{11} + ( 2 \zeta_{24} - 4 \zeta_{24}^{3} + 2 \zeta_{24}^{4} - 4 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{13} + ( -2 + 4 \zeta_{24}^{4} ) q^{17} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{19} + ( -2 \zeta_{24} - 2 \zeta_{24}^{7} ) q^{23} + ( -4 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{25} + ( 2 + 4 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{29} + ( \zeta_{24} + 5 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 5 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{31} + ( 4 \zeta_{24} - 6 \zeta_{24}^{2} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 3 \zeta_{24}^{6} ) q^{35} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{37} + ( 8 + 2 \zeta_{24} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{7} ) q^{41} + ( -4 \zeta_{24} - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{43} + ( 2 \zeta_{24}^{2} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 4 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{47} + ( 2 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{49} + ( 5 - 3 \zeta_{24} + 3 \zeta_{24}^{3} - 10 \zeta_{24}^{4} + 3 \zeta_{24}^{5} ) q^{53} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 7 \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{55} + ( -4 \zeta_{24} - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{59} + ( -4 + 4 \zeta_{24}^{4} ) q^{61} + ( -6 + 8 \zeta_{24}^{3} - 6 \zeta_{24}^{4} + 8 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{65} + ( 2 \zeta_{24} - 10 \zeta_{24}^{2} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 10 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{67} + ( -2 \zeta_{24} + 12 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 6 \zeta_{24}^{6} ) q^{71} + ( -3 + 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{73} + ( -10 + 5 \zeta_{24} + 5 \zeta_{24}^{4} - 5 \zeta_{24}^{7} ) q^{77} + 2 \zeta_{24}^{2} q^{79} + ( \zeta_{24}^{2} + 10 \zeta_{24}^{3} - 10 \zeta_{24}^{5} - 2 \zeta_{24}^{6} - 10 \zeta_{24}^{7} ) q^{83} + ( -2 \zeta_{24} + 4 \zeta_{24}^{3} - 6 \zeta_{24}^{4} + 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{85} + ( -2 - 6 \zeta_{24} + 6 \zeta_{24}^{3} + 4 \zeta_{24}^{4} + 6 \zeta_{24}^{5} ) q^{89} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 14 \zeta_{24}^{6} - 8 \zeta_{24}^{7} ) q^{91} + ( -6 \zeta_{24} + 4 \zeta_{24}^{2} + 4 \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{95} + ( 5 - 5 \zeta_{24}^{4} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 12q^{5} + O(q^{10})$$ $$8q - 12q^{5} + 8q^{13} + 24q^{29} + 48q^{41} - 16q^{61} - 72q^{65} - 24q^{73} - 60q^{77} - 24q^{85} + 20q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$325$$ $$1217$$ $$2431$$ $$\chi(n)$$ $$1$$ $$\zeta_{24}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
863.1
 −0.258819 − 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i 0.258819 + 0.965926i −0.258819 + 0.965926i −0.965926 − 0.258819i 0.965926 + 0.258819i 0.258819 − 0.965926i
0 0 0 −2.72474 1.57313i 0 −2.98735 + 1.72474i 0 0 0
863.2 0 0 0 −2.72474 1.57313i 0 2.98735 1.72474i 0 0 0
863.3 0 0 0 −0.275255 0.158919i 0 −1.25529 + 0.724745i 0 0 0
863.4 0 0 0 −0.275255 0.158919i 0 1.25529 0.724745i 0 0 0
1727.1 0 0 0 −2.72474 + 1.57313i 0 −2.98735 1.72474i 0 0 0
1727.2 0 0 0 −2.72474 + 1.57313i 0 2.98735 + 1.72474i 0 0 0
1727.3 0 0 0 −0.275255 + 0.158919i 0 −1.25529 0.724745i 0 0 0
1727.4 0 0 0 −0.275255 + 0.158919i 0 1.25529 + 0.724745i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1727.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.d odd 6 1 inner
36.h even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2592.2.s.a 8
3.b odd 2 1 2592.2.s.h 8
4.b odd 2 1 inner 2592.2.s.a 8
9.c even 3 1 864.2.c.a 8
9.c even 3 1 2592.2.s.h 8
9.d odd 6 1 864.2.c.a 8
9.d odd 6 1 inner 2592.2.s.a 8
12.b even 2 1 2592.2.s.h 8
36.f odd 6 1 864.2.c.a 8
36.f odd 6 1 2592.2.s.h 8
36.h even 6 1 864.2.c.a 8
36.h even 6 1 inner 2592.2.s.a 8
72.j odd 6 1 1728.2.c.g 8
72.l even 6 1 1728.2.c.g 8
72.n even 6 1 1728.2.c.g 8
72.p odd 6 1 1728.2.c.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.c.a 8 9.c even 3 1
864.2.c.a 8 9.d odd 6 1
864.2.c.a 8 36.f odd 6 1
864.2.c.a 8 36.h even 6 1
1728.2.c.g 8 72.j odd 6 1
1728.2.c.g 8 72.l even 6 1
1728.2.c.g 8 72.n even 6 1
1728.2.c.g 8 72.p odd 6 1
2592.2.s.a 8 1.a even 1 1 trivial
2592.2.s.a 8 4.b odd 2 1 inner
2592.2.s.a 8 9.d odd 6 1 inner
2592.2.s.a 8 36.h even 6 1 inner
2592.2.s.h 8 3.b odd 2 1
2592.2.s.h 8 9.c even 3 1
2592.2.s.h 8 12.b even 2 1
2592.2.s.h 8 36.f odd 6 1

## Hecke kernels

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

 $$T_{5}^{4} + 6 T_{5}^{3} + 13 T_{5}^{2} + 6 T_{5} + 1$$ $$T_{7}^{8} - 14 T_{7}^{6} + 171 T_{7}^{4} - 350 T_{7}^{2} + 625$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 1 + 6 T + 13 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$7$ $$625 - 350 T^{2} + 171 T^{4} - 14 T^{6} + T^{8}$$
$11$ $$625 + 550 T^{2} + 459 T^{4} + 22 T^{6} + T^{8}$$
$13$ $$( 400 + 80 T + 36 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$17$ $$( 12 + T^{2} )^{4}$$
$19$ $$( 24 + T^{2} )^{4}$$
$23$ $$( 64 + 8 T^{2} + T^{4} )^{2}$$
$29$ $$( 400 + 240 T + 28 T^{2} - 12 T^{3} + T^{4} )^{2}$$
$31$ $$130321 - 22382 T^{2} + 3483 T^{4} - 62 T^{6} + T^{8}$$
$37$ $$( -24 + T^{2} )^{4}$$
$41$ $$( 1600 - 960 T + 232 T^{2} - 24 T^{3} + T^{4} )^{2}$$
$43$ $$160000 - 22400 T^{2} + 2736 T^{4} - 56 T^{6} + T^{8}$$
$47$ $$160000 + 35200 T^{2} + 7344 T^{4} + 88 T^{6} + T^{8}$$
$53$ $$( 3249 + 186 T^{2} + T^{4} )^{2}$$
$59$ $$160000 + 35200 T^{2} + 7344 T^{4} + 88 T^{6} + T^{8}$$
$61$ $$( 16 + 4 T + T^{2} )^{4}$$
$67$ $$33362176 - 1432448 T^{2} + 55728 T^{4} - 248 T^{6} + T^{8}$$
$71$ $$( 10000 - 232 T^{2} + T^{4} )^{2}$$
$73$ $$( -15 + 6 T + T^{2} )^{4}$$
$79$ $$( 16 - 4 T^{2} + T^{4} )^{2}$$
$83$ $$1506138481 + 15756454 T^{2} + 126027 T^{4} + 406 T^{6} + T^{8}$$
$89$ $$( 3600 + 168 T^{2} + T^{4} )^{2}$$
$97$ $$( 25 - 5 T + T^{2} )^{4}$$