# Properties

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

# 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{5} + (2 \zeta_{6} - 2) q^{7}+O(q^{10})$$ q + z * q^5 + (2*z - 2) * q^7 $$q + \zeta_{6} q^{5} + (2 \zeta_{6} - 2) q^{7} + ( - 2 \zeta_{6} + 2) q^{11} - \zeta_{6} q^{13} - 3 q^{17} + 2 q^{19} + 6 \zeta_{6} q^{23} + ( - 4 \zeta_{6} + 4) q^{25} + ( - \zeta_{6} + 1) q^{29} + 8 \zeta_{6} q^{31} - 2 q^{35} + q^{37} - 2 \zeta_{6} q^{41} + (10 \zeta_{6} - 10) q^{43} + ( - 4 \zeta_{6} + 4) q^{47} + 3 \zeta_{6} q^{49} + 10 q^{53} + 2 q^{55} + 4 \zeta_{6} q^{59} + (9 \zeta_{6} - 9) q^{61} + ( - \zeta_{6} + 1) q^{65} + 14 \zeta_{6} q^{67} - 10 q^{71} - 9 q^{73} + 4 \zeta_{6} q^{77} + (10 \zeta_{6} - 10) q^{79} + ( - 12 \zeta_{6} + 12) q^{83} - 3 \zeta_{6} q^{85} - 11 q^{89} + 2 q^{91} + 2 \zeta_{6} q^{95} + ( - 2 \zeta_{6} + 2) q^{97} +O(q^{100})$$ q + z * q^5 + (2*z - 2) * q^7 + (-2*z + 2) * q^11 - z * q^13 - 3 * q^17 + 2 * q^19 + 6*z * q^23 + (-4*z + 4) * q^25 + (-z + 1) * q^29 + 8*z * q^31 - 2 * q^35 + q^37 - 2*z * q^41 + (10*z - 10) * q^43 + (-4*z + 4) * q^47 + 3*z * q^49 + 10 * q^53 + 2 * q^55 + 4*z * q^59 + (9*z - 9) * q^61 + (-z + 1) * q^65 + 14*z * q^67 - 10 * q^71 - 9 * q^73 + 4*z * q^77 + (10*z - 10) * q^79 + (-12*z + 12) * q^83 - 3*z * q^85 - 11 * q^89 + 2 * q^91 + 2*z * q^95 + (-2*z + 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{5} - 2 q^{7}+O(q^{10})$$ 2 * q + q^5 - 2 * q^7 $$2 q + q^{5} - 2 q^{7} + 2 q^{11} - q^{13} - 6 q^{17} + 4 q^{19} + 6 q^{23} + 4 q^{25} + q^{29} + 8 q^{31} - 4 q^{35} + 2 q^{37} - 2 q^{41} - 10 q^{43} + 4 q^{47} + 3 q^{49} + 20 q^{53} + 4 q^{55} + 4 q^{59} - 9 q^{61} + q^{65} + 14 q^{67} - 20 q^{71} - 18 q^{73} + 4 q^{77} - 10 q^{79} + 12 q^{83} - 3 q^{85} - 22 q^{89} + 4 q^{91} + 2 q^{95} + 2 q^{97}+O(q^{100})$$ 2 * q + q^5 - 2 * q^7 + 2 * q^11 - q^13 - 6 * q^17 + 4 * q^19 + 6 * q^23 + 4 * q^25 + q^29 + 8 * q^31 - 4 * q^35 + 2 * q^37 - 2 * q^41 - 10 * q^43 + 4 * q^47 + 3 * q^49 + 20 * q^53 + 4 * q^55 + 4 * q^59 - 9 * q^61 + q^65 + 14 * q^67 - 20 * q^71 - 18 * q^73 + 4 * q^77 - 10 * q^79 + 12 * q^83 - 3 * q^85 - 22 * q^89 + 4 * q^91 + 2 * q^95 + 2 * 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 0.500000 + 0.866025i 0 −1.00000 + 1.73205i 0 0 0
1729.1 0 0 0 0.500000 0.866025i 0 −1.00000 1.73205i 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.n 2
3.b odd 2 1 2592.2.i.j 2
4.b odd 2 1 2592.2.i.o 2
9.c even 3 1 2592.2.a.d yes 1
9.c even 3 1 inner 2592.2.i.n 2
9.d odd 6 1 2592.2.a.f yes 1
9.d odd 6 1 2592.2.i.j 2
12.b even 2 1 2592.2.i.k 2
36.f odd 6 1 2592.2.a.c 1
36.f odd 6 1 2592.2.i.o 2
36.h even 6 1 2592.2.a.e yes 1
36.h even 6 1 2592.2.i.k 2
72.j odd 6 1 5184.2.a.m 1
72.l even 6 1 5184.2.a.j 1
72.n even 6 1 5184.2.a.w 1
72.p odd 6 1 5184.2.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2592.2.a.c 1 36.f odd 6 1
2592.2.a.d yes 1 9.c even 3 1
2592.2.a.e yes 1 36.h even 6 1
2592.2.a.f yes 1 9.d odd 6 1
2592.2.i.j 2 3.b odd 2 1
2592.2.i.j 2 9.d odd 6 1
2592.2.i.k 2 12.b even 2 1
2592.2.i.k 2 36.h even 6 1
2592.2.i.n 2 1.a even 1 1 trivial
2592.2.i.n 2 9.c even 3 1 inner
2592.2.i.o 2 4.b odd 2 1
2592.2.i.o 2 36.f odd 6 1
5184.2.a.j 1 72.l even 6 1
5184.2.a.m 1 72.j odd 6 1
5184.2.a.t 1 72.p odd 6 1
5184.2.a.w 1 72.n even 6 1

## Hecke kernels

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

 $$T_{5}^{2} - T_{5} + 1$$ T5^2 - T5 + 1 $$T_{7}^{2} + 2T_{7} + 4$$ T7^2 + 2*T7 + 4 $$T_{11}^{2} - 2T_{11} + 4$$ T11^2 - 2*T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2} + 2T + 4$$
$11$ $$T^{2} - 2T + 4$$
$13$ $$T^{2} + T + 1$$
$17$ $$(T + 3)^{2}$$
$19$ $$(T - 2)^{2}$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$T^{2} - T + 1$$
$31$ $$T^{2} - 8T + 64$$
$37$ $$(T - 1)^{2}$$
$41$ $$T^{2} + 2T + 4$$
$43$ $$T^{2} + 10T + 100$$
$47$ $$T^{2} - 4T + 16$$
$53$ $$(T - 10)^{2}$$
$59$ $$T^{2} - 4T + 16$$
$61$ $$T^{2} + 9T + 81$$
$67$ $$T^{2} - 14T + 196$$
$71$ $$(T + 10)^{2}$$
$73$ $$(T + 9)^{2}$$
$79$ $$T^{2} + 10T + 100$$
$83$ $$T^{2} - 12T + 144$$
$89$ $$(T + 11)^{2}$$
$97$ $$T^{2} - 2T + 4$$