# Properties

 Label 2592.2.i.r Level $2592$ Weight $2$ Character orbit 2592.i Analytic conductor $20.697$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{5} + ( -3 + 3 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + 2 \zeta_{6} q^{5} + ( -3 + 3 \zeta_{6} ) q^{7} + ( 6 - 6 \zeta_{6} ) q^{11} + 3 \zeta_{6} q^{13} + 2 q^{17} -3 q^{19} + 6 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{25} + ( -8 + 8 \zeta_{6} ) q^{29} -6 q^{35} + 7 q^{37} + 8 \zeta_{6} q^{41} + ( 12 - 12 \zeta_{6} ) q^{43} + ( 6 - 6 \zeta_{6} ) q^{47} -2 \zeta_{6} q^{49} -4 q^{53} + 12 q^{55} + 6 \zeta_{6} q^{59} + ( 1 - \zeta_{6} ) q^{61} + ( -6 + 6 \zeta_{6} ) q^{65} + 3 \zeta_{6} q^{67} -12 q^{71} -15 q^{73} + 18 \zeta_{6} q^{77} + ( -9 + 9 \zeta_{6} ) q^{79} + ( -12 + 12 \zeta_{6} ) q^{83} + 4 \zeta_{6} q^{85} + 10 q^{89} -9 q^{91} -6 \zeta_{6} q^{95} + ( -9 + 9 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} - 3 q^{7} + O(q^{10})$$ $$2 q + 2 q^{5} - 3 q^{7} + 6 q^{11} + 3 q^{13} + 4 q^{17} - 6 q^{19} + 6 q^{23} + q^{25} - 8 q^{29} - 12 q^{35} + 14 q^{37} + 8 q^{41} + 12 q^{43} + 6 q^{47} - 2 q^{49} - 8 q^{53} + 24 q^{55} + 6 q^{59} + q^{61} - 6 q^{65} + 3 q^{67} - 24 q^{71} - 30 q^{73} + 18 q^{77} - 9 q^{79} - 12 q^{83} + 4 q^{85} + 20 q^{89} - 18 q^{91} - 6 q^{95} - 9 q^{97} + O(q^{100})$$

## 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 1.00000 + 1.73205i 0 −1.50000 + 2.59808i 0 0 0
1729.1 0 0 0 1.00000 1.73205i 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.r 2
3.b odd 2 1 2592.2.i.c 2
4.b odd 2 1 2592.2.i.v 2
9.c even 3 1 864.2.a.d yes 1
9.c even 3 1 inner 2592.2.i.r 2
9.d odd 6 1 864.2.a.l yes 1
9.d odd 6 1 2592.2.i.c 2
12.b even 2 1 2592.2.i.g 2
36.f odd 6 1 864.2.a.a 1
36.f odd 6 1 2592.2.i.v 2
36.h even 6 1 864.2.a.i yes 1
36.h even 6 1 2592.2.i.g 2
72.j odd 6 1 1728.2.a.h 1
72.l even 6 1 1728.2.a.e 1
72.n even 6 1 1728.2.a.x 1
72.p odd 6 1 1728.2.a.u 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.a.a 1 36.f odd 6 1
864.2.a.d yes 1 9.c even 3 1
864.2.a.i yes 1 36.h even 6 1
864.2.a.l yes 1 9.d odd 6 1
1728.2.a.e 1 72.l even 6 1
1728.2.a.h 1 72.j odd 6 1
1728.2.a.u 1 72.p odd 6 1
1728.2.a.x 1 72.n even 6 1
2592.2.i.c 2 3.b odd 2 1
2592.2.i.c 2 9.d odd 6 1
2592.2.i.g 2 12.b even 2 1
2592.2.i.g 2 36.h even 6 1
2592.2.i.r 2 1.a even 1 1 trivial
2592.2.i.r 2 9.c even 3 1 inner
2592.2.i.v 2 4.b odd 2 1
2592.2.i.v 2 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 $$S_{2}^{\mathrm{new}}(2592, [\chi])$$:

 $$T_{5}^{2} - 2 T_{5} + 4$$ $$T_{7}^{2} + 3 T_{7} + 9$$ $$T_{11}^{2} - 6 T_{11} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$4 - 2 T + T^{2}$$
$7$ $$9 + 3 T + T^{2}$$
$11$ $$36 - 6 T + T^{2}$$
$13$ $$9 - 3 T + T^{2}$$
$17$ $$( -2 + T )^{2}$$
$19$ $$( 3 + T )^{2}$$
$23$ $$36 - 6 T + T^{2}$$
$29$ $$64 + 8 T + T^{2}$$
$31$ $$T^{2}$$
$37$ $$( -7 + T )^{2}$$
$41$ $$64 - 8 T + T^{2}$$
$43$ $$144 - 12 T + T^{2}$$
$47$ $$36 - 6 T + T^{2}$$
$53$ $$( 4 + T )^{2}$$
$59$ $$36 - 6 T + T^{2}$$
$61$ $$1 - T + T^{2}$$
$67$ $$9 - 3 T + T^{2}$$
$71$ $$( 12 + T )^{2}$$
$73$ $$( 15 + T )^{2}$$
$79$ $$81 + 9 T + T^{2}$$
$83$ $$144 + 12 T + T^{2}$$
$89$ $$( -10 + T )^{2}$$
$97$ $$81 + 9 T + T^{2}$$