Properties

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

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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: \(\Z[a_1, \ldots, a_{25}]\)
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.

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

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

Hecke kernels

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

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