# Properties

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

# 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: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: no (minimal twist has level 96) 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.

 $$f(q)$$ $$=$$ $$q + ( 2 \zeta_{24} - 2 \zeta_{24}^{7} ) q^{5} -2 \zeta_{24}^{2} q^{7} +O(q^{10})$$ $$q + ( 2 \zeta_{24} - 2 \zeta_{24}^{7} ) q^{5} -2 \zeta_{24}^{2} q^{7} + ( -2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{11} + 2 \zeta_{24}^{4} q^{13} -6 \zeta_{24}^{6} q^{19} + ( -4 \zeta_{24} - 4 \zeta_{24}^{7} ) q^{23} + ( 3 - 3 \zeta_{24}^{4} ) q^{25} + ( 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{29} + ( 2 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{31} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{35} + 6 q^{37} + ( 4 \zeta_{24} - 4 \zeta_{24}^{7} ) q^{41} -2 \zeta_{24}^{2} q^{43} + ( 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{47} -3 \zeta_{24}^{4} q^{49} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{53} + 8 \zeta_{24}^{6} q^{55} + ( -2 \zeta_{24} - 2 \zeta_{24}^{7} ) q^{59} + ( 2 - 2 \zeta_{24}^{4} ) q^{61} + ( 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{65} + ( 2 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{67} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{71} -6 q^{73} + ( 4 \zeta_{24} - 4 \zeta_{24}^{7} ) q^{77} + 14 \zeta_{24}^{2} q^{79} + ( 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{83} + ( 12 \zeta_{24} - 12 \zeta_{24}^{3} - 12 \zeta_{24}^{5} ) q^{89} -4 \zeta_{24}^{6} q^{91} + ( -12 \zeta_{24} - 12 \zeta_{24}^{7} ) q^{95} + ( -10 + 10 \zeta_{24}^{4} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + O(q^{10})$$ $$8 q + 8 q^{13} + 12 q^{25} + 48 q^{37} - 12 q^{49} + 8 q^{61} - 48 q^{73} - 40 q^{97} + O(q^{100})$$

## Character values

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

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

## 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
 −0.965926 + 0.258819i −0.258819 − 0.965926i 0.965926 − 0.258819i 0.258819 + 0.965926i −0.965926 − 0.258819i −0.258819 + 0.965926i 0.965926 + 0.258819i 0.258819 − 0.965926i
0 0 0 −2.44949 1.41421i 0 −1.73205 + 1.00000i 0 0 0
863.2 0 0 0 −2.44949 1.41421i 0 1.73205 1.00000i 0 0 0
863.3 0 0 0 2.44949 + 1.41421i 0 −1.73205 + 1.00000i 0 0 0
863.4 0 0 0 2.44949 + 1.41421i 0 1.73205 1.00000i 0 0 0
1727.1 0 0 0 −2.44949 + 1.41421i 0 −1.73205 1.00000i 0 0 0
1727.2 0 0 0 −2.44949 + 1.41421i 0 1.73205 + 1.00000i 0 0 0
1727.3 0 0 0 2.44949 1.41421i 0 −1.73205 1.00000i 0 0 0
1727.4 0 0 0 2.44949 1.41421i 0 1.73205 + 1.00000i 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
4.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
12.b even 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.e 8
3.b odd 2 1 inner 2592.2.s.e 8
4.b odd 2 1 inner 2592.2.s.e 8
9.c even 3 1 96.2.c.a 4
9.c even 3 1 inner 2592.2.s.e 8
9.d odd 6 1 96.2.c.a 4
9.d odd 6 1 inner 2592.2.s.e 8
12.b even 2 1 inner 2592.2.s.e 8
36.f odd 6 1 96.2.c.a 4
36.f odd 6 1 inner 2592.2.s.e 8
36.h even 6 1 96.2.c.a 4
36.h even 6 1 inner 2592.2.s.e 8
45.h odd 6 1 2400.2.h.c 4
45.j even 6 1 2400.2.h.c 4
45.k odd 12 1 2400.2.o.a 4
45.k odd 12 1 2400.2.o.h 4
45.l even 12 1 2400.2.o.a 4
45.l even 12 1 2400.2.o.h 4
72.j odd 6 1 192.2.c.b 4
72.l even 6 1 192.2.c.b 4
72.n even 6 1 192.2.c.b 4
72.p odd 6 1 192.2.c.b 4
144.u even 12 1 768.2.f.a 4
144.u even 12 1 768.2.f.g 4
144.v odd 12 1 768.2.f.a 4
144.v odd 12 1 768.2.f.g 4
144.w odd 12 1 768.2.f.a 4
144.w odd 12 1 768.2.f.g 4
144.x even 12 1 768.2.f.a 4
144.x even 12 1 768.2.f.g 4
180.n even 6 1 2400.2.h.c 4
180.p odd 6 1 2400.2.h.c 4
180.v odd 12 1 2400.2.o.a 4
180.v odd 12 1 2400.2.o.h 4
180.x even 12 1 2400.2.o.a 4
180.x even 12 1 2400.2.o.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.2.c.a 4 9.c even 3 1
96.2.c.a 4 9.d odd 6 1
96.2.c.a 4 36.f odd 6 1
96.2.c.a 4 36.h even 6 1
192.2.c.b 4 72.j odd 6 1
192.2.c.b 4 72.l even 6 1
192.2.c.b 4 72.n even 6 1
192.2.c.b 4 72.p odd 6 1
768.2.f.a 4 144.u even 12 1
768.2.f.a 4 144.v odd 12 1
768.2.f.a 4 144.w odd 12 1
768.2.f.a 4 144.x even 12 1
768.2.f.g 4 144.u even 12 1
768.2.f.g 4 144.v odd 12 1
768.2.f.g 4 144.w odd 12 1
768.2.f.g 4 144.x even 12 1
2400.2.h.c 4 45.h odd 6 1
2400.2.h.c 4 45.j even 6 1
2400.2.h.c 4 180.n even 6 1
2400.2.h.c 4 180.p odd 6 1
2400.2.o.a 4 45.k odd 12 1
2400.2.o.a 4 45.l even 12 1
2400.2.o.a 4 180.v odd 12 1
2400.2.o.a 4 180.x even 12 1
2400.2.o.h 4 45.k odd 12 1
2400.2.o.h 4 45.l even 12 1
2400.2.o.h 4 180.v odd 12 1
2400.2.o.h 4 180.x even 12 1
2592.2.s.e 8 1.a even 1 1 trivial
2592.2.s.e 8 3.b odd 2 1 inner
2592.2.s.e 8 4.b odd 2 1 inner
2592.2.s.e 8 9.c even 3 1 inner
2592.2.s.e 8 9.d odd 6 1 inner
2592.2.s.e 8 12.b even 2 1 inner
2592.2.s.e 8 36.f odd 6 1 inner
2592.2.s.e 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 $$S_{2}^{\mathrm{new}}(2592, [\chi])$$:

 $$T_{5}^{4} - 8 T_{5}^{2} + 64$$ $$T_{7}^{4} - 4 T_{7}^{2} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 64 - 8 T^{2} + T^{4} )^{2}$$
$7$ $$( 16 - 4 T^{2} + T^{4} )^{2}$$
$11$ $$( 64 + 8 T^{2} + T^{4} )^{2}$$
$13$ $$( 4 - 2 T + T^{2} )^{4}$$
$17$ $$T^{8}$$
$19$ $$( 36 + T^{2} )^{4}$$
$23$ $$( 1024 + 32 T^{2} + T^{4} )^{2}$$
$29$ $$( 64 - 8 T^{2} + T^{4} )^{2}$$
$31$ $$( 16 - 4 T^{2} + T^{4} )^{2}$$
$37$ $$( -6 + T )^{8}$$
$41$ $$( 1024 - 32 T^{2} + T^{4} )^{2}$$
$43$ $$( 16 - 4 T^{2} + T^{4} )^{2}$$
$47$ $$( 16384 + 128 T^{2} + T^{4} )^{2}$$
$53$ $$( 72 + T^{2} )^{4}$$
$59$ $$( 64 + 8 T^{2} + T^{4} )^{2}$$
$61$ $$( 4 - 2 T + T^{2} )^{4}$$
$67$ $$( 16 - 4 T^{2} + T^{4} )^{2}$$
$71$ $$( -32 + T^{2} )^{4}$$
$73$ $$( 6 + T )^{8}$$
$79$ $$( 38416 - 196 T^{2} + T^{4} )^{2}$$
$83$ $$( 64 + 8 T^{2} + T^{4} )^{2}$$
$89$ $$( 288 + T^{2} )^{4}$$
$97$ $$( 100 + 10 T + T^{2} )^{4}$$