Properties

Label 1440.4.a.i
Level $1440$
Weight $4$
Character orbit 1440.a
Self dual yes
Analytic conductor $84.963$
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] = [1440,4,Mod(1,1440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1440, 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("1440.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1440.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: \(84.9627504083\)
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)
 
\(f(q)\) \(=\) \( q - 5 q^{5} + 30 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{5} + 30 q^{7} + 50 q^{11} - 88 q^{13} + 74 q^{17} + 140 q^{19} + 80 q^{23} + 25 q^{25} - 234 q^{29} - 150 q^{35} + 116 q^{37} - 72 q^{41} + 280 q^{43} + 120 q^{47} + 557 q^{49} - 498 q^{53} - 250 q^{55} - 870 q^{59} + 650 q^{61} + 440 q^{65} + 420 q^{67} + 1020 q^{71} - 322 q^{73} + 1500 q^{77} + 160 q^{79} - 980 q^{83} - 370 q^{85} - 1124 q^{89} - 2640 q^{91} - 700 q^{95} + 1114 q^{97}+O(q^{100}) \) 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 1440.4.a.i yes 1
3.b odd 2 1 1440.4.a.r yes 1
4.b odd 2 1 1440.4.a.b 1
12.b even 2 1 1440.4.a.k yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1440.4.a.b 1 4.b odd 2 1
1440.4.a.i yes 1 1.a even 1 1 trivial
1440.4.a.k yes 1 12.b even 2 1
1440.4.a.r yes 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1440))\):

\( T_{7} - 30 \) Copy content Toggle raw display
\( T_{11} - 50 \) Copy content Toggle raw display
\( T_{17} - 74 \) 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
$13$ \( T + 88 \) Copy content Toggle raw display
$17$ \( T - 74 \) Copy content Toggle raw display
$19$ \( T - 140 \) Copy content Toggle raw display
$23$ \( T - 80 \) Copy content Toggle raw display
$29$ \( T + 234 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 116 \) Copy content Toggle raw display
$41$ \( T + 72 \) Copy content Toggle raw display
$43$ \( T - 280 \) Copy content Toggle raw display
$47$ \( T - 120 \) Copy content Toggle raw display
$53$ \( T + 498 \) Copy content Toggle raw display
$59$ \( T + 870 \) Copy content Toggle raw display
$61$ \( T - 650 \) Copy content Toggle raw display
$67$ \( T - 420 \) Copy content Toggle raw display
$71$ \( T - 1020 \) Copy content Toggle raw display
$73$ \( T + 322 \) Copy content Toggle raw display
$79$ \( T - 160 \) Copy content Toggle raw display
$83$ \( T + 980 \) Copy content Toggle raw display
$89$ \( T + 1124 \) Copy content Toggle raw display
$97$ \( T - 1114 \) Copy content Toggle raw display
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