# Properties

 Label 2592.2.s.g 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: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{8}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 2 \zeta_{24} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{5} + ( 2 - 2 \zeta_{24}^{2} - \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{7} +O(q^{10})$$ $$q + ( 2 \zeta_{24} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{5} + ( 2 - 2 \zeta_{24}^{2} - \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{7} + ( 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{11} + ( -1 + 2 \zeta_{24}^{2} + \zeta_{24}^{4} - 4 \zeta_{24}^{6} ) q^{13} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{17} + ( 1 - 2 \zeta_{24}^{4} ) q^{19} + ( 6 \zeta_{24} - 4 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{23} + ( -4 \zeta_{24}^{2} + 3 \zeta_{24}^{4} - 4 \zeta_{24}^{6} ) q^{25} + ( -4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{29} + ( 2 - 4 \zeta_{24}^{2} + 2 \zeta_{24}^{4} ) q^{31} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} - 8 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{35} + ( -3 - 4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{37} + ( 4 \zeta_{24} - 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{41} + ( 4 + 4 \zeta_{24}^{2} - 2 \zeta_{24}^{4} - 4 \zeta_{24}^{6} ) q^{43} + ( 4 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{47} + ( -4 \zeta_{24}^{2} + 8 \zeta_{24}^{6} ) q^{49} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 8 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{53} + 4 \zeta_{24}^{6} q^{55} + ( -2 \zeta_{24} + 8 \zeta_{24}^{3} - 6 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{59} + ( 6 \zeta_{24}^{2} - \zeta_{24}^{4} + 6 \zeta_{24}^{6} ) q^{61} + ( 4 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{65} + ( 1 + 8 \zeta_{24}^{2} + \zeta_{24}^{4} ) q^{67} + ( 8 \zeta_{24} + 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} ) q^{71} + ( 3 + 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{73} + ( -2 \zeta_{24} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{77} + ( -6 - 10 \zeta_{24}^{2} + 3 \zeta_{24}^{4} + 10 \zeta_{24}^{6} ) q^{79} + ( 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{83} + ( 4 \zeta_{24}^{2} - 8 \zeta_{24}^{6} ) q^{85} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{89} + ( -5 + 10 \zeta_{24}^{4} - 8 \zeta_{24}^{6} ) q^{91} + ( -2 \zeta_{24} + 4 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{95} -7 \zeta_{24}^{4} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 12 q^{7} + O(q^{10})$$ $$8 q + 12 q^{7} - 4 q^{13} + 12 q^{25} + 24 q^{31} - 24 q^{37} + 24 q^{43} - 4 q^{61} + 12 q^{67} + 24 q^{73} - 36 q^{79} - 28 q^{97} + O(q^{100})$$

## Character values

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

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

## 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 −3.34607 1.93185i 0 3.23205 1.86603i 0 0 0
863.2 0 0 0 −0.896575 0.517638i 0 −0.232051 + 0.133975i 0 0 0
863.3 0 0 0 0.896575 + 0.517638i 0 −0.232051 + 0.133975i 0 0 0
863.4 0 0 0 3.34607 + 1.93185i 0 3.23205 1.86603i 0 0 0
1727.1 0 0 0 −3.34607 + 1.93185i 0 3.23205 + 1.86603i 0 0 0
1727.2 0 0 0 −0.896575 + 0.517638i 0 −0.232051 0.133975i 0 0 0
1727.3 0 0 0 0.896575 0.517638i 0 −0.232051 0.133975i 0 0 0
1727.4 0 0 0 3.34607 1.93185i 0 3.23205 + 1.86603i 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
3.b odd 2 1 inner
36.f 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.g 8
3.b odd 2 1 inner 2592.2.s.g 8
4.b odd 2 1 2592.2.s.c 8
9.c even 3 1 864.2.c.b 8
9.c even 3 1 2592.2.s.c 8
9.d odd 6 1 864.2.c.b 8
9.d odd 6 1 2592.2.s.c 8
12.b even 2 1 2592.2.s.c 8
36.f odd 6 1 864.2.c.b 8
36.f odd 6 1 inner 2592.2.s.g 8
36.h even 6 1 864.2.c.b 8
36.h even 6 1 inner 2592.2.s.g 8
72.j odd 6 1 1728.2.c.f 8
72.l even 6 1 1728.2.c.f 8
72.n even 6 1 1728.2.c.f 8
72.p odd 6 1 1728.2.c.f 8

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

## Hecke kernels

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

 $$T_{5}^{8} - 16 T_{5}^{6} + 240 T_{5}^{4} - 256 T_{5}^{2} + 256$$ $$T_{7}^{4} - 6 T_{7}^{3} + 11 T_{7}^{2} + 6 T_{7} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$256 - 256 T^{2} + 240 T^{4} - 16 T^{6} + T^{8}$$
$7$ $$( 1 + 6 T + 11 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$11$ $$256 + 256 T^{2} + 240 T^{4} + 16 T^{6} + T^{8}$$
$13$ $$( 121 - 22 T + 15 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$17$ $$( 144 + 48 T^{2} + T^{4} )^{2}$$
$19$ $$( 3 + T^{2} )^{4}$$
$23$ $$7311616 + 302848 T^{2} + 9840 T^{4} + 112 T^{6} + T^{8}$$
$29$ $$65536 - 16384 T^{2} + 3840 T^{4} - 64 T^{6} + T^{8}$$
$31$ $$( 16 + 48 T + 44 T^{2} - 12 T^{3} + T^{4} )^{2}$$
$37$ $$( -3 + 6 T + T^{2} )^{4}$$
$41$ $$65536 - 16384 T^{2} + 3840 T^{4} - 64 T^{6} + T^{8}$$
$43$ $$( 16 + 48 T + 44 T^{2} - 12 T^{3} + T^{4} )^{2}$$
$47$ $$3748096 + 216832 T^{2} + 10608 T^{4} + 112 T^{6} + T^{8}$$
$53$ $$( 2304 + 192 T^{2} + T^{4} )^{2}$$
$59$ $$71639296 + 1760512 T^{2} + 34800 T^{4} + 208 T^{6} + T^{8}$$
$61$ $$( 11449 - 214 T + 111 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$67$ $$( 3721 + 366 T - 49 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$71$ $$( -128 + T^{2} )^{4}$$
$73$ $$( -39 - 6 T + T^{2} )^{4}$$
$79$ $$( 5329 - 1314 T + 35 T^{2} + 18 T^{3} + T^{4} )^{2}$$
$83$ $$65536 + 16384 T^{2} + 3840 T^{4} + 64 T^{6} + T^{8}$$
$89$ $$( 144 + 48 T^{2} + T^{4} )^{2}$$
$97$ $$( 49 + 7 T + T^{2} )^{4}$$