Properties

Label 2592.2.a.m
Level $2592$
Weight $2$
Character orbit 2592.a
Self dual yes
Analytic conductor $20.697$
Analytic rank $1$
Dimension $2$
CM no
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(1,2592)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2592, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2592.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2592 = 2^{5} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2592.a (trivial)

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: yes
Analytic conductor: \(20.6972242039\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = 2\sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{5} + \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{5} + \beta q^{7} - \beta q^{11} + 3 q^{13} - 5 q^{17} - \beta q^{19} - \beta q^{23} - 4 q^{25} - 5 q^{29} - \beta q^{35} - 5 q^{37} + 2 q^{41} - \beta q^{43} + 2 \beta q^{47} + 17 q^{49} - 2 q^{53} + \beta q^{55} + 2 \beta q^{59} - 13 q^{61} - 3 q^{65} - \beta q^{67} + \beta q^{71} + 3 q^{73} - 24 q^{77} - 3 \beta q^{79} + 2 \beta q^{83} + 5 q^{85} - 13 q^{89} + 3 \beta q^{91} + \beta q^{95} - 6 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} + 6 q^{13} - 10 q^{17} - 8 q^{25} - 10 q^{29} - 10 q^{37} + 4 q^{41} + 34 q^{49} - 4 q^{53} - 26 q^{61} - 6 q^{65} + 6 q^{73} - 48 q^{77} + 10 q^{85} - 26 q^{89} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.44949
2.44949
0 0 0 −1.00000 0 −4.89898 0 0 0
1.2 0 0 0 −1.00000 0 4.89898 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2592.2.a.m 2
3.b odd 2 1 2592.2.a.r yes 2
4.b odd 2 1 inner 2592.2.a.m 2
8.b even 2 1 5184.2.a.bv 2
8.d odd 2 1 5184.2.a.bv 2
9.c even 3 2 2592.2.i.bd 4
9.d odd 6 2 2592.2.i.z 4
12.b even 2 1 2592.2.a.r yes 2
24.f even 2 1 5184.2.a.bk 2
24.h odd 2 1 5184.2.a.bk 2
36.f odd 6 2 2592.2.i.bd 4
36.h even 6 2 2592.2.i.z 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2592.2.a.m 2 1.a even 1 1 trivial
2592.2.a.m 2 4.b odd 2 1 inner
2592.2.a.r yes 2 3.b odd 2 1
2592.2.a.r yes 2 12.b even 2 1
2592.2.i.z 4 9.d odd 6 2
2592.2.i.z 4 36.h even 6 2
2592.2.i.bd 4 9.c even 3 2
2592.2.i.bd 4 36.f odd 6 2
5184.2.a.bk 2 24.f even 2 1
5184.2.a.bk 2 24.h odd 2 1
5184.2.a.bv 2 8.b even 2 1
5184.2.a.bv 2 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2592))\):

\( T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} - 24 \) Copy content Toggle raw display
\( T_{11}^{2} - 24 \) 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
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 24 \) Copy content Toggle raw display
$11$ \( T^{2} - 24 \) Copy content Toggle raw display
$13$ \( (T - 3)^{2} \) Copy content Toggle raw display
$17$ \( (T + 5)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 24 \) Copy content Toggle raw display
$23$ \( T^{2} - 24 \) Copy content Toggle raw display
$29$ \( (T + 5)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T + 5)^{2} \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 24 \) Copy content Toggle raw display
$47$ \( T^{2} - 96 \) Copy content Toggle raw display
$53$ \( (T + 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 96 \) Copy content Toggle raw display
$61$ \( (T + 13)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 24 \) Copy content Toggle raw display
$71$ \( T^{2} - 24 \) Copy content Toggle raw display
$73$ \( (T - 3)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 216 \) Copy content Toggle raw display
$83$ \( T^{2} - 96 \) Copy content Toggle raw display
$89$ \( (T + 13)^{2} \) Copy content Toggle raw display
$97$ \( (T + 6)^{2} \) Copy content Toggle raw display
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