# Properties

 Label 2592.2.i.x Level $2592$ Weight $2$ Character orbit 2592.i Analytic conductor $20.697$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2592,2,Mod(865,2592)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2592, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2592.865");

S:= CuspForms(chi, 2);

N := Newforms(S);

 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

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$20.6972242039$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 288) Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q + 4 \zeta_{6} q^{5}+O(q^{10})$$ q + 4*z * q^5 $$q + 4 \zeta_{6} q^{5} + 6 \zeta_{6} q^{13} - 8 q^{17} + (11 \zeta_{6} - 11) q^{25} + (4 \zeta_{6} - 4) q^{29} - 2 q^{37} - 8 \zeta_{6} q^{41} + 7 \zeta_{6} q^{49} + 4 q^{53} + ( - 10 \zeta_{6} + 10) q^{61} + (24 \zeta_{6} - 24) q^{65} + 6 q^{73} - 32 \zeta_{6} q^{85} - 16 q^{89} + ( - 18 \zeta_{6} + 18) q^{97} +O(q^{100})$$ q + 4*z * q^5 + 6*z * q^13 - 8 * q^17 + (11*z - 11) * q^25 + (4*z - 4) * q^29 - 2 * q^37 - 8*z * q^41 + 7*z * q^49 + 4 * q^53 + (-10*z + 10) * q^61 + (24*z - 24) * q^65 + 6 * q^73 - 32*z * q^85 - 16 * q^89 + (-18*z + 18) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{5}+O(q^{10})$$ 2 * q + 4 * q^5 $$2 q + 4 q^{5} + 6 q^{13} - 16 q^{17} - 11 q^{25} - 4 q^{29} - 4 q^{37} - 8 q^{41} + 7 q^{49} + 8 q^{53} + 10 q^{61} - 24 q^{65} + 12 q^{73} - 32 q^{85} - 32 q^{89} + 18 q^{97}+O(q^{100})$$ 2 * q + 4 * q^5 + 6 * q^13 - 16 * q^17 - 11 * q^25 - 4 * q^29 - 4 * q^37 - 8 * q^41 + 7 * q^49 + 8 * q^53 + 10 * q^61 - 24 * q^65 + 12 * q^73 - 32 * q^85 - 32 * q^89 + 18 * q^97

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 2.00000 + 3.46410i 0 0 0 0 0
1729.1 0 0 0 2.00000 3.46410i 0 0 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
9.c even 3 1 inner
36.f odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2592.2.i.x 2
3.b odd 2 1 2592.2.i.a 2
4.b odd 2 1 CM 2592.2.i.x 2
9.c even 3 1 288.2.a.a 1
9.c even 3 1 inner 2592.2.i.x 2
9.d odd 6 1 288.2.a.e yes 1
9.d odd 6 1 2592.2.i.a 2
12.b even 2 1 2592.2.i.a 2
36.f odd 6 1 288.2.a.a 1
36.f odd 6 1 inner 2592.2.i.x 2
36.h even 6 1 288.2.a.e yes 1
36.h even 6 1 2592.2.i.a 2
45.h odd 6 1 7200.2.a.be 1
45.j even 6 1 7200.2.a.bf 1
45.k odd 12 2 7200.2.f.n 2
45.l even 12 2 7200.2.f.q 2
72.j odd 6 1 576.2.a.a 1
72.l even 6 1 576.2.a.a 1
72.n even 6 1 576.2.a.i 1
72.p odd 6 1 576.2.a.i 1
144.u even 12 2 2304.2.d.l 2
144.v odd 12 2 2304.2.d.h 2
144.w odd 12 2 2304.2.d.l 2
144.x even 12 2 2304.2.d.h 2
180.n even 6 1 7200.2.a.be 1
180.p odd 6 1 7200.2.a.bf 1
180.v odd 12 2 7200.2.f.q 2
180.x even 12 2 7200.2.f.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.a.a 1 9.c even 3 1
288.2.a.a 1 36.f odd 6 1
288.2.a.e yes 1 9.d odd 6 1
288.2.a.e yes 1 36.h even 6 1
576.2.a.a 1 72.j odd 6 1
576.2.a.a 1 72.l even 6 1
576.2.a.i 1 72.n even 6 1
576.2.a.i 1 72.p odd 6 1
2304.2.d.h 2 144.v odd 12 2
2304.2.d.h 2 144.x even 12 2
2304.2.d.l 2 144.u even 12 2
2304.2.d.l 2 144.w odd 12 2
2592.2.i.a 2 3.b odd 2 1
2592.2.i.a 2 9.d odd 6 1
2592.2.i.a 2 12.b even 2 1
2592.2.i.a 2 36.h even 6 1
2592.2.i.x 2 1.a even 1 1 trivial
2592.2.i.x 2 4.b odd 2 1 CM
2592.2.i.x 2 9.c even 3 1 inner
2592.2.i.x 2 36.f odd 6 1 inner
7200.2.a.be 1 45.h odd 6 1
7200.2.a.be 1 180.n even 6 1
7200.2.a.bf 1 45.j even 6 1
7200.2.a.bf 1 180.p odd 6 1
7200.2.f.n 2 45.k odd 12 2
7200.2.f.n 2 180.x even 12 2
7200.2.f.q 2 45.l even 12 2
7200.2.f.q 2 180.v odd 12 2

## Hecke kernels

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

 $$T_{5}^{2} - 4T_{5} + 16$$ T5^2 - 4*T5 + 16 $$T_{7}$$ T7 $$T_{11}$$ T11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 4T + 16$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 6T + 36$$
$17$ $$(T + 8)^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 4T + 16$$
$31$ $$T^{2}$$
$37$ $$(T + 2)^{2}$$
$41$ $$T^{2} + 8T + 64$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$(T - 4)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 10T + 100$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$(T - 6)^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$(T + 16)^{2}$$
$97$ $$T^{2} - 18T + 324$$