Properties

Label 1440.4.a.t
Level $1440$
Weight $4$
Character orbit 1440.a
Self dual yes
Analytic conductor $84.963$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 5 q^{5} + (5 \beta - 4) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{5} + (5 \beta - 4) q^{7} + (6 \beta + 32) q^{11} + (8 \beta + 6) q^{13} + (24 \beta - 2) q^{17} + (8 \beta - 104) q^{19} + ( - 11 \beta + 60) q^{23} + 25 q^{25} + (16 \beta + 146) q^{29} + (6 \beta - 88) q^{31} + ( - 25 \beta + 20) q^{35} + (16 \beta - 178) q^{37} + (40 \beta - 50) q^{41} + ( - 37 \beta + 188) q^{43} + ( - 13 \beta - 140) q^{47} + ( - 40 \beta + 273) q^{49} + ( - 24 \beta - 158) q^{53} + ( - 30 \beta - 160) q^{55} + ( - 20 \beta + 360) q^{59} + ( - 32 \beta - 634) q^{61} + ( - 40 \beta - 30) q^{65} + ( - 47 \beta + 372) q^{67} + (218 \beta - 24) q^{71} + ( - 120 \beta - 470) q^{73} + (136 \beta + 592) q^{77} + (172 \beta - 16) q^{79} + ( - 47 \beta - 796) q^{83} + ( - 120 \beta + 10) q^{85} + ( - 48 \beta + 390) q^{89} + ( - 2 \beta + 936) q^{91} + ( - 40 \beta + 520) q^{95} + ( - 40 \beta + 610) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 10 q^{5} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 10 q^{5} - 8 q^{7} + 64 q^{11} + 12 q^{13} - 4 q^{17} - 208 q^{19} + 120 q^{23} + 50 q^{25} + 292 q^{29} - 176 q^{31} + 40 q^{35} - 356 q^{37} - 100 q^{41} + 376 q^{43} - 280 q^{47} + 546 q^{49} - 316 q^{53} - 320 q^{55} + 720 q^{59} - 1268 q^{61} - 60 q^{65} + 744 q^{67} - 48 q^{71} - 940 q^{73} + 1184 q^{77} - 32 q^{79} - 1592 q^{83} + 20 q^{85} + 780 q^{89} + 1872 q^{91} + 1040 q^{95} + 1220 q^{97}+O(q^{100}) \) 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
−2.44949
2.44949
0 0 0 −5.00000 0 −28.4949 0 0 0
1.2 0 0 0 −5.00000 0 20.4949 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.t 2
3.b odd 2 1 160.4.a.c 2
4.b odd 2 1 1440.4.a.x 2
12.b even 2 1 160.4.a.g yes 2
15.d odd 2 1 800.4.a.s 2
15.e even 4 2 800.4.c.i 4
24.f even 2 1 320.4.a.o 2
24.h odd 2 1 320.4.a.s 2
48.i odd 4 2 1280.4.d.x 4
48.k even 4 2 1280.4.d.q 4
60.h even 2 1 800.4.a.m 2
60.l odd 4 2 800.4.c.k 4
120.i odd 2 1 1600.4.a.cd 2
120.m even 2 1 1600.4.a.cn 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.c 2 3.b odd 2 1
160.4.a.g yes 2 12.b even 2 1
320.4.a.o 2 24.f even 2 1
320.4.a.s 2 24.h odd 2 1
800.4.a.m 2 60.h even 2 1
800.4.a.s 2 15.d odd 2 1
800.4.c.i 4 15.e even 4 2
800.4.c.k 4 60.l odd 4 2
1280.4.d.q 4 48.k even 4 2
1280.4.d.x 4 48.i odd 4 2
1440.4.a.t 2 1.a even 1 1 trivial
1440.4.a.x 2 4.b odd 2 1
1600.4.a.cd 2 120.i odd 2 1
1600.4.a.cn 2 120.m even 2 1

Hecke kernels

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

\( T_{7}^{2} + 8T_{7} - 584 \) Copy content Toggle raw display
\( T_{11}^{2} - 64T_{11} + 160 \) Copy content Toggle raw display
\( T_{17}^{2} + 4T_{17} - 13820 \) 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
$7$ \( T^{2} + 8T - 584 \) Copy content Toggle raw display
$11$ \( T^{2} - 64T + 160 \) Copy content Toggle raw display
$13$ \( T^{2} - 12T - 1500 \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 13820 \) Copy content Toggle raw display
$19$ \( T^{2} + 208T + 9280 \) Copy content Toggle raw display
$23$ \( T^{2} - 120T + 696 \) Copy content Toggle raw display
$29$ \( T^{2} - 292T + 15172 \) Copy content Toggle raw display
$31$ \( T^{2} + 176T + 6880 \) Copy content Toggle raw display
$37$ \( T^{2} + 356T + 25540 \) Copy content Toggle raw display
$41$ \( T^{2} + 100T - 35900 \) Copy content Toggle raw display
$43$ \( T^{2} - 376T + 2488 \) Copy content Toggle raw display
$47$ \( T^{2} + 280T + 15544 \) Copy content Toggle raw display
$53$ \( T^{2} + 316T + 11140 \) Copy content Toggle raw display
$59$ \( T^{2} - 720T + 120000 \) Copy content Toggle raw display
$61$ \( T^{2} + 1268 T + 377380 \) Copy content Toggle raw display
$67$ \( T^{2} - 744T + 85368 \) Copy content Toggle raw display
$71$ \( T^{2} + 48T - 1140000 \) Copy content Toggle raw display
$73$ \( T^{2} + 940T - 124700 \) Copy content Toggle raw display
$79$ \( T^{2} + 32T - 709760 \) Copy content Toggle raw display
$83$ \( T^{2} + 1592 T + 580600 \) Copy content Toggle raw display
$89$ \( T^{2} - 780T + 96804 \) Copy content Toggle raw display
$97$ \( T^{2} - 1220 T + 333700 \) Copy content Toggle raw display
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