Properties

Label 2592.2.a.v
Level $2592$
Weight $2$
Character orbit 2592.a
Self dual yes
Analytic conductor $20.697$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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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: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} + ( - \beta_{2} + \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{5} + ( - \beta_{2} + \beta_1) q^{7} + ( - \beta_{3} - 1) q^{11} + (\beta_{3} - 2) q^{13} - \beta_1 q^{17} + ( - \beta_{2} - 3 \beta_1) q^{19} + (\beta_{3} - 3) q^{23} + 2 q^{25} + (\beta_{2} - 2 \beta_1) q^{29} + ( - 2 \beta_{2} + 2 \beta_1) q^{31} + (\beta_{3} - 7) q^{35} + ( - \beta_{3} - 2) q^{37} + ( - 2 \beta_{2} - 2 \beta_1) q^{41} + (\beta_{2} + 3 \beta_1) q^{43} - 8 q^{47} + ( - 2 \beta_{3} + 3) q^{49} + (2 \beta_{2} + 2 \beta_1) q^{53} + ( - \beta_{2} - 7 \beta_1) q^{55} - 8 q^{59} + ( - \beta_{3} - 6) q^{61} + ( - 2 \beta_{2} + 7 \beta_1) q^{65} + (\beta_{2} + 3 \beta_1) q^{67} + ( - \beta_{3} - 5) q^{71} + (2 \beta_{3} + 3) q^{73} + ( - 2 \beta_{2} + 6 \beta_1) q^{77} + (\beta_{2} + 7 \beta_1) q^{79} + (2 \beta_{3} - 6) q^{83} - \beta_{3} q^{85} + ( - 4 \beta_{2} + \beta_1) q^{89} + (5 \beta_{2} - 9 \beta_1) q^{91} + ( - 3 \beta_{3} - 7) q^{95} + (2 \beta_{3} + 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{11} - 8 q^{13} - 12 q^{23} + 8 q^{25} - 28 q^{35} - 8 q^{37} - 32 q^{47} + 12 q^{49} - 32 q^{59} - 24 q^{61} - 20 q^{71} + 12 q^{73} - 24 q^{83} - 28 q^{95} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} - 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{3} + 6\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} - 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} + 3\beta_1 \) 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
−2.18890
−0.456850
0.456850
2.18890
0 0 0 −2.64575 0 0.913701 0 0 0
1.2 0 0 0 −2.64575 0 4.37780 0 0 0
1.3 0 0 0 2.64575 0 −4.37780 0 0 0
1.4 0 0 0 2.64575 0 −0.913701 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
12.b even 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.v 4
3.b odd 2 1 2592.2.a.w yes 4
4.b odd 2 1 2592.2.a.w yes 4
8.b even 2 1 5184.2.a.ce 4
8.d odd 2 1 5184.2.a.cd 4
9.c even 3 2 2592.2.i.bh 8
9.d odd 6 2 2592.2.i.bg 8
12.b even 2 1 inner 2592.2.a.v 4
24.f even 2 1 5184.2.a.ce 4
24.h odd 2 1 5184.2.a.cd 4
36.f odd 6 2 2592.2.i.bg 8
36.h even 6 2 2592.2.i.bh 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2592.2.a.v 4 1.a even 1 1 trivial
2592.2.a.v 4 12.b even 2 1 inner
2592.2.a.w yes 4 3.b odd 2 1
2592.2.a.w yes 4 4.b odd 2 1
2592.2.i.bg 8 9.d odd 6 2
2592.2.i.bg 8 36.f odd 6 2
2592.2.i.bh 8 9.c even 3 2
2592.2.i.bh 8 36.h even 6 2
5184.2.a.cd 4 8.d odd 2 1
5184.2.a.cd 4 24.h odd 2 1
5184.2.a.ce 4 8.b even 2 1
5184.2.a.ce 4 24.f even 2 1

Hecke kernels

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

\( T_{5}^{2} - 7 \) Copy content Toggle raw display
\( T_{7}^{4} - 20T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} - 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 7)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 20T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T - 20)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 4 T - 17)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 68T^{2} + 400 \) Copy content Toggle raw display
$23$ \( (T^{2} + 6 T - 12)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 38T^{2} + 25 \) Copy content Toggle raw display
$31$ \( T^{4} - 80T^{2} + 256 \) Copy content Toggle raw display
$37$ \( (T^{2} + 4 T - 17)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 80T^{2} + 256 \) Copy content Toggle raw display
$43$ \( T^{4} - 68T^{2} + 400 \) Copy content Toggle raw display
$47$ \( (T + 8)^{4} \) Copy content Toggle raw display
$53$ \( T^{4} - 80T^{2} + 256 \) Copy content Toggle raw display
$59$ \( (T + 8)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 12 T + 15)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 68T^{2} + 400 \) Copy content Toggle raw display
$71$ \( (T^{2} + 10 T + 4)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 6 T - 75)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 308 T^{2} + 19600 \) Copy content Toggle raw display
$83$ \( (T^{2} + 12 T - 48)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 230 T^{2} + 11881 \) Copy content Toggle raw display
$97$ \( (T^{2} - 8 T - 68)^{2} \) Copy content Toggle raw display
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