Properties

Label 720.4.a.o
Level $720$
Weight $4$
Character orbit 720.a
Self dual yes
Analytic conductor $42.481$
Analytic rank $1$
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] = [720,4,Mod(1,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("720.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 720.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: \(42.4813752041\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
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)
 
\(f(q)\) \(=\) \( q - 5 q^{5} + 30 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{5} + 30 q^{7} - 50 q^{11} - 20 q^{13} - 10 q^{17} + 44 q^{19} - 120 q^{23} + 25 q^{25} - 50 q^{29} - 108 q^{31} - 150 q^{35} - 40 q^{37} + 400 q^{41} - 280 q^{43} + 280 q^{47} + 557 q^{49} - 610 q^{53} + 250 q^{55} - 50 q^{59} - 518 q^{61} + 100 q^{65} + 180 q^{67} - 700 q^{71} - 410 q^{73} - 1500 q^{77} + 516 q^{79} - 660 q^{83} + 50 q^{85} - 1500 q^{89} - 600 q^{91} - 220 q^{95} - 1630 q^{97}+O(q^{100}) \) 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 30.0000 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 720.4.a.o 1
3.b odd 2 1 720.4.a.bc 1
4.b odd 2 1 45.4.a.e yes 1
12.b even 2 1 45.4.a.a 1
20.d odd 2 1 225.4.a.a 1
20.e even 4 2 225.4.b.b 2
28.d even 2 1 2205.4.a.t 1
36.f odd 6 2 405.4.e.b 2
36.h even 6 2 405.4.e.n 2
60.h even 2 1 225.4.a.h 1
60.l odd 4 2 225.4.b.a 2
84.h odd 2 1 2205.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.a.a 1 12.b even 2 1
45.4.a.e yes 1 4.b odd 2 1
225.4.a.a 1 20.d odd 2 1
225.4.a.h 1 60.h even 2 1
225.4.b.a 2 60.l odd 4 2
225.4.b.b 2 20.e even 4 2
405.4.e.b 2 36.f odd 6 2
405.4.e.n 2 36.h even 6 2
720.4.a.o 1 1.a even 1 1 trivial
720.4.a.bc 1 3.b odd 2 1
2205.4.a.a 1 84.h odd 2 1
2205.4.a.t 1 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(720))\):

\( T_{7} - 30 \) Copy content Toggle raw display
\( T_{11} + 50 \) 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
$7$ \( T - 30 \) Copy content Toggle raw display
$11$ \( T + 50 \) Copy content Toggle raw display
$13$ \( T + 20 \) Copy content Toggle raw display
$17$ \( T + 10 \) Copy content Toggle raw display
$19$ \( T - 44 \) Copy content Toggle raw display
$23$ \( T + 120 \) Copy content Toggle raw display
$29$ \( T + 50 \) Copy content Toggle raw display
$31$ \( T + 108 \) Copy content Toggle raw display
$37$ \( T + 40 \) Copy content Toggle raw display
$41$ \( T - 400 \) Copy content Toggle raw display
$43$ \( T + 280 \) Copy content Toggle raw display
$47$ \( T - 280 \) Copy content Toggle raw display
$53$ \( T + 610 \) Copy content Toggle raw display
$59$ \( T + 50 \) Copy content Toggle raw display
$61$ \( T + 518 \) Copy content Toggle raw display
$67$ \( T - 180 \) Copy content Toggle raw display
$71$ \( T + 700 \) Copy content Toggle raw display
$73$ \( T + 410 \) Copy content Toggle raw display
$79$ \( T - 516 \) Copy content Toggle raw display
$83$ \( T + 660 \) Copy content Toggle raw display
$89$ \( T + 1500 \) Copy content Toggle raw display
$97$ \( T + 1630 \) Copy content Toggle raw display
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