Properties

Label 2592.2.i.k
Level $2592$
Weight $2$
Character orbit 2592.i
Analytic conductor $20.697$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{5} + ( - 2 \zeta_{6} + 2) q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6} q^{5} + ( - 2 \zeta_{6} + 2) q^{7} + ( - 2 \zeta_{6} + 2) q^{11} - \zeta_{6} q^{13} + 3 q^{17} - 2 q^{19} + 6 \zeta_{6} q^{23} + ( - 4 \zeta_{6} + 4) q^{25} + (\zeta_{6} - 1) q^{29} - 8 \zeta_{6} q^{31} - 2 q^{35} + q^{37} + 2 \zeta_{6} q^{41} + ( - 10 \zeta_{6} + 10) q^{43} + ( - 4 \zeta_{6} + 4) q^{47} + 3 \zeta_{6} q^{49} - 10 q^{53} - 2 q^{55} + 4 \zeta_{6} q^{59} + (9 \zeta_{6} - 9) q^{61} + (\zeta_{6} - 1) q^{65} - 14 \zeta_{6} q^{67} - 10 q^{71} - 9 q^{73} - 4 \zeta_{6} q^{77} + ( - 10 \zeta_{6} + 10) q^{79} + ( - 12 \zeta_{6} + 12) q^{83} - 3 \zeta_{6} q^{85} + 11 q^{89} - 2 q^{91} + 2 \zeta_{6} q^{95} + ( - 2 \zeta_{6} + 2) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{5} + 2 q^{7} + 2 q^{11} - q^{13} + 6 q^{17} - 4 q^{19} + 6 q^{23} + 4 q^{25} - q^{29} - 8 q^{31} - 4 q^{35} + 2 q^{37} + 2 q^{41} + 10 q^{43} + 4 q^{47} + 3 q^{49} - 20 q^{53} - 4 q^{55} + 4 q^{59} - 9 q^{61} - q^{65} - 14 q^{67} - 20 q^{71} - 18 q^{73} - 4 q^{77} + 10 q^{79} + 12 q^{83} - 3 q^{85} + 22 q^{89} - 4 q^{91} + 2 q^{95} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
0.500000 + 0.866025i
0.500000 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:

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

Hecke kernels

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

\( T_{5}^{2} + T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} - 2T_{7} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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