Properties

Label 1440.4.a.bc
Level $1440$
Weight $4$
Character orbit 1440.a
Self dual yes
Analytic conductor $84.963$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{65}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{65}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 5 q^{5} - \beta q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q + 5 q^{5} - \beta q^{7} - \beta q^{11} - 8 q^{13} - 26 q^{17} + 6 \beta q^{19} + 8 \beta q^{23} + 25 q^{25} - 54 q^{29} + 8 \beta q^{31} - 5 \beta q^{35} - 124 q^{37} - 152 q^{41} + 12 \beta q^{43} + 4 \beta q^{47} - 83 q^{49} - 78 q^{53} - 5 \beta q^{55} - 37 \beta q^{59} - 470 q^{61} - 40 q^{65} - 46 \beta q^{67} - 14 \beta q^{71} - 162 q^{73} + 260 q^{77} - 8 \beta q^{79} + 74 \beta q^{83} - 130 q^{85} - 124 q^{89} + 8 \beta q^{91} + 30 \beta q^{95} - 646 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 10 q^{5} - 16 q^{13} - 52 q^{17} + 50 q^{25} - 108 q^{29} - 248 q^{37} - 304 q^{41} - 166 q^{49} - 156 q^{53} - 940 q^{61} - 80 q^{65} - 324 q^{73} + 520 q^{77} - 260 q^{85} - 248 q^{89} - 1292 q^{97}+O(q^{100}) \) 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
4.53113
−3.53113
0 0 0 5.00000 0 −16.1245 0 0 0
1.2 0 0 0 5.00000 0 16.1245 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1440.4.a.bc yes 2
3.b odd 2 1 1440.4.a.u 2
4.b odd 2 1 inner 1440.4.a.bc yes 2
12.b even 2 1 1440.4.a.u 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1440.4.a.u 2 3.b odd 2 1
1440.4.a.u 2 12.b even 2 1
1440.4.a.bc yes 2 1.a even 1 1 trivial
1440.4.a.bc yes 2 4.b odd 2 1 inner

Hecke kernels

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

\( T_{7}^{2} - 260 \) Copy content Toggle raw display
\( T_{11}^{2} - 260 \) Copy content Toggle raw display
\( T_{17} + 26 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 260 \) Copy content Toggle raw display
$11$ \( T^{2} - 260 \) Copy content Toggle raw display
$13$ \( (T + 8)^{2} \) Copy content Toggle raw display
$17$ \( (T + 26)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 9360 \) Copy content Toggle raw display
$23$ \( T^{2} - 16640 \) Copy content Toggle raw display
$29$ \( (T + 54)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 16640 \) Copy content Toggle raw display
$37$ \( (T + 124)^{2} \) Copy content Toggle raw display
$41$ \( (T + 152)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 37440 \) Copy content Toggle raw display
$47$ \( T^{2} - 4160 \) Copy content Toggle raw display
$53$ \( (T + 78)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 355940 \) Copy content Toggle raw display
$61$ \( (T + 470)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 550160 \) Copy content Toggle raw display
$71$ \( T^{2} - 50960 \) Copy content Toggle raw display
$73$ \( (T + 162)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 16640 \) Copy content Toggle raw display
$83$ \( T^{2} - 1423760 \) Copy content Toggle raw display
$89$ \( (T + 124)^{2} \) Copy content Toggle raw display
$97$ \( (T + 646)^{2} \) Copy content Toggle raw display
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