Properties

Label 540.4.a.a
Level $540$
Weight $4$
Character orbit 540.a
Self dual yes
Analytic conductor $31.861$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [540,4,Mod(1,540)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(540, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("540.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 540 = 2^{2} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 540.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: \(31.8610314031\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
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)
 
\(f(q)\) \(=\) \( q - 5 q^{5} - 22 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{5} - 22 q^{7} - 9 q^{11} + 17 q^{13} - 75 q^{17} - 4 q^{19} + 183 q^{23} + 25 q^{25} + 129 q^{29} - 187 q^{31} + 110 q^{35} - 34 q^{37} + 264 q^{41} + 443 q^{43} + 609 q^{47} + 141 q^{49} - 228 q^{53} + 45 q^{55} + 60 q^{59} - 454 q^{61} - 85 q^{65} - 244 q^{67} + 444 q^{71} + 398 q^{73} + 198 q^{77} - 349 q^{79} + 1038 q^{83} + 375 q^{85} + 852 q^{89} - 374 q^{91} + 20 q^{95} + 914 q^{97}+O(q^{100}) \) 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
0
0 0 0 −5.00000 0 −22.0000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 540.4.a.a 1
3.b odd 2 1 540.4.a.c yes 1
4.b odd 2 1 2160.4.a.h 1
9.c even 3 2 1620.4.i.k 2
9.d odd 6 2 1620.4.i.e 2
12.b even 2 1 2160.4.a.s 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.4.a.a 1 1.a even 1 1 trivial
540.4.a.c yes 1 3.b odd 2 1
1620.4.i.e 2 9.d odd 6 2
1620.4.i.k 2 9.c even 3 2
2160.4.a.h 1 4.b odd 2 1
2160.4.a.s 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(540))\):

\( T_{7} + 22 \) Copy content Toggle raw display
\( T_{11} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 5 \) Copy content Toggle raw display
$7$ \( T + 22 \) Copy content Toggle raw display
$11$ \( T + 9 \) Copy content Toggle raw display
$13$ \( T - 17 \) Copy content Toggle raw display
$17$ \( T + 75 \) Copy content Toggle raw display
$19$ \( T + 4 \) Copy content Toggle raw display
$23$ \( T - 183 \) Copy content Toggle raw display
$29$ \( T - 129 \) Copy content Toggle raw display
$31$ \( T + 187 \) Copy content Toggle raw display
$37$ \( T + 34 \) Copy content Toggle raw display
$41$ \( T - 264 \) Copy content Toggle raw display
$43$ \( T - 443 \) Copy content Toggle raw display
$47$ \( T - 609 \) Copy content Toggle raw display
$53$ \( T + 228 \) Copy content Toggle raw display
$59$ \( T - 60 \) Copy content Toggle raw display
$61$ \( T + 454 \) Copy content Toggle raw display
$67$ \( T + 244 \) Copy content Toggle raw display
$71$ \( T - 444 \) Copy content Toggle raw display
$73$ \( T - 398 \) Copy content Toggle raw display
$79$ \( T + 349 \) Copy content Toggle raw display
$83$ \( T - 1038 \) Copy content Toggle raw display
$89$ \( T - 852 \) Copy content Toggle raw display
$97$ \( T - 914 \) Copy content Toggle raw display
show more
show less