Properties

Label 1440.4.a.n
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: no (minimal twist has level 160)
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)
 
\(f(q)\) \(=\) \( q + 5 q^{5} - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 q^{5} - 6 q^{7} + 60 q^{11} + 50 q^{13} + 30 q^{17} - 40 q^{19} + 178 q^{23} + 25 q^{25} - 166 q^{29} - 20 q^{31} - 30 q^{35} + 10 q^{37} + 250 q^{41} - 142 q^{43} + 214 q^{47} - 307 q^{49} - 490 q^{53} + 300 q^{55} - 800 q^{59} + 250 q^{61} + 250 q^{65} + 774 q^{67} + 100 q^{71} - 230 q^{73} - 360 q^{77} + 1320 q^{79} + 982 q^{83} + 150 q^{85} - 874 q^{89} - 300 q^{91} - 200 q^{95} - 310 q^{97}+O(q^{100}) \) 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 −6.00000 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 1440.4.a.n 1
3.b odd 2 1 160.4.a.b yes 1
4.b odd 2 1 1440.4.a.o 1
12.b even 2 1 160.4.a.a 1
15.d odd 2 1 800.4.a.d 1
15.e even 4 2 800.4.c.e 2
24.f even 2 1 320.4.a.i 1
24.h odd 2 1 320.4.a.f 1
48.i odd 4 2 1280.4.d.k 2
48.k even 4 2 1280.4.d.f 2
60.h even 2 1 800.4.a.h 1
60.l odd 4 2 800.4.c.f 2
120.i odd 2 1 1600.4.a.bj 1
120.m even 2 1 1600.4.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.a 1 12.b even 2 1
160.4.a.b yes 1 3.b odd 2 1
320.4.a.f 1 24.h odd 2 1
320.4.a.i 1 24.f even 2 1
800.4.a.d 1 15.d odd 2 1
800.4.a.h 1 60.h even 2 1
800.4.c.e 2 15.e even 4 2
800.4.c.f 2 60.l odd 4 2
1280.4.d.f 2 48.k even 4 2
1280.4.d.k 2 48.i odd 4 2
1440.4.a.n 1 1.a even 1 1 trivial
1440.4.a.o 1 4.b odd 2 1
1600.4.a.r 1 120.m even 2 1
1600.4.a.bj 1 120.i odd 2 1

Hecke kernels

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

\( T_{7} + 6 \) Copy content Toggle raw display
\( T_{11} - 60 \) Copy content Toggle raw display
\( T_{17} - 30 \) 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
$7$ \( T + 6 \) Copy content Toggle raw display
$11$ \( T - 60 \) Copy content Toggle raw display
$13$ \( T - 50 \) Copy content Toggle raw display
$17$ \( T - 30 \) Copy content Toggle raw display
$19$ \( T + 40 \) Copy content Toggle raw display
$23$ \( T - 178 \) Copy content Toggle raw display
$29$ \( T + 166 \) Copy content Toggle raw display
$31$ \( T + 20 \) Copy content Toggle raw display
$37$ \( T - 10 \) Copy content Toggle raw display
$41$ \( T - 250 \) Copy content Toggle raw display
$43$ \( T + 142 \) Copy content Toggle raw display
$47$ \( T - 214 \) Copy content Toggle raw display
$53$ \( T + 490 \) Copy content Toggle raw display
$59$ \( T + 800 \) Copy content Toggle raw display
$61$ \( T - 250 \) Copy content Toggle raw display
$67$ \( T - 774 \) Copy content Toggle raw display
$71$ \( T - 100 \) Copy content Toggle raw display
$73$ \( T + 230 \) Copy content Toggle raw display
$79$ \( T - 1320 \) Copy content Toggle raw display
$83$ \( T - 982 \) Copy content Toggle raw display
$89$ \( T + 874 \) Copy content Toggle raw display
$97$ \( T + 310 \) Copy content Toggle raw display
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