Properties

Label 1080.2.a.f
Level $1080$
Weight $2$
Character orbit 1080.a
Self dual yes
Analytic conductor $8.624$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1080,2,Mod(1,1080)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1080, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1080.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1080.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: \(8.62384341830\)
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)
 
\(f(q)\) \(=\) \( q - q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{5} + 2 q^{7} - q^{11} + q^{13} + q^{17} + 4 q^{19} - q^{23} + q^{25} + 5 q^{29} + q^{31} - 2 q^{35} + 6 q^{37} + 7 q^{43} - 7 q^{47} - 3 q^{49} + 12 q^{53} + q^{55} + 4 q^{59} + 10 q^{61} - q^{65} - 4 q^{67} - 12 q^{71} + 6 q^{73} - 2 q^{77} + 15 q^{79} - 2 q^{83} - q^{85} + 12 q^{89} + 2 q^{91} - 4 q^{95} + 10 q^{97}+O(q^{100}) \) 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 −1.00000 0 2.00000 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 1080.2.a.f 1
3.b odd 2 1 1080.2.a.k yes 1
4.b odd 2 1 2160.2.a.d 1
5.b even 2 1 5400.2.a.m 1
5.c odd 4 2 5400.2.f.m 2
8.b even 2 1 8640.2.a.cb 1
8.d odd 2 1 8640.2.a.bk 1
9.c even 3 2 3240.2.q.o 2
9.d odd 6 2 3240.2.q.c 2
12.b even 2 1 2160.2.a.n 1
15.d odd 2 1 5400.2.a.n 1
15.e even 4 2 5400.2.f.p 2
24.f even 2 1 8640.2.a.i 1
24.h odd 2 1 8640.2.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.a.f 1 1.a even 1 1 trivial
1080.2.a.k yes 1 3.b odd 2 1
2160.2.a.d 1 4.b odd 2 1
2160.2.a.n 1 12.b even 2 1
3240.2.q.c 2 9.d odd 6 2
3240.2.q.o 2 9.c even 3 2
5400.2.a.m 1 5.b even 2 1
5400.2.a.n 1 15.d odd 2 1
5400.2.f.m 2 5.c odd 4 2
5400.2.f.p 2 15.e even 4 2
8640.2.a.i 1 24.f even 2 1
8640.2.a.v 1 24.h odd 2 1
8640.2.a.bk 1 8.d odd 2 1
8640.2.a.cb 1 8.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1080))\):

\( T_{7} - 2 \) Copy content Toggle raw display
\( T_{11} + 1 \) Copy content Toggle raw display
\( T_{17} - 1 \) 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
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T - 2 \) Copy content Toggle raw display
$11$ \( T + 1 \) Copy content Toggle raw display
$13$ \( T - 1 \) Copy content Toggle raw display
$17$ \( T - 1 \) Copy content Toggle raw display
$19$ \( T - 4 \) Copy content Toggle raw display
$23$ \( T + 1 \) Copy content Toggle raw display
$29$ \( T - 5 \) Copy content Toggle raw display
$31$ \( T - 1 \) Copy content Toggle raw display
$37$ \( T - 6 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T - 7 \) Copy content Toggle raw display
$47$ \( T + 7 \) Copy content Toggle raw display
$53$ \( T - 12 \) Copy content Toggle raw display
$59$ \( T - 4 \) Copy content Toggle raw display
$61$ \( T - 10 \) Copy content Toggle raw display
$67$ \( T + 4 \) Copy content Toggle raw display
$71$ \( T + 12 \) Copy content Toggle raw display
$73$ \( T - 6 \) Copy content Toggle raw display
$79$ \( T - 15 \) Copy content Toggle raw display
$83$ \( T + 2 \) Copy content Toggle raw display
$89$ \( T - 12 \) Copy content Toggle raw display
$97$ \( T - 10 \) Copy content Toggle raw display
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