Properties

Label 1440.4.a.bk
Level $1440$
Weight $4$
Character orbit 1440.a
Self dual yes
Analytic conductor $84.963$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.16773.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 17x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 5 q^{5} + ( - \beta_1 + 5) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 q^{5} + ( - \beta_1 + 5) q^{7} + (\beta_{2} + 2 \beta_1 + 7) q^{11} + (\beta_{2} - 3 \beta_1 + 4) q^{13} + (\beta_{2} + 5 \beta_1 + 10) q^{17} + (3 \beta_{2} - \beta_1) q^{19} + ( - 3 \beta_{2} - \beta_1 + 58) q^{23} + 25 q^{25} + (6 \beta_{2} - 2 \beta_1 - 30) q^{29} + ( - 5 \beta_{2} - 7 \beta_1 + 30) q^{31} + ( - 5 \beta_1 + 25) q^{35} + ( - 9 \beta_{2} - 13 \beta_1 + 96) q^{37} - 8 \beta_{2} q^{41} + ( - 2 \beta_{2} + 22 \beta_1 + 160) q^{43} + (9 \beta_{2} - 17 \beta_1 + 102) q^{47} + (4 \beta_{2} - 12 \beta_1 - 131) q^{49} + ( - 12 \beta_{2} - 12 \beta_1 + 50) q^{53} + (5 \beta_{2} + 10 \beta_1 + 35) q^{55} + ( - 9 \beta_{2} + 32 \beta_1 + 103) q^{59} + (18 \beta_{2} + 26 \beta_1 + 114) q^{61} + (5 \beta_{2} - 15 \beta_1 + 20) q^{65} + ( - 4 \beta_{2} - 10 \beta_1 + 270) q^{67} + ( - 8 \beta_{2} - 6 \beta_1 + 166) q^{71} + ( - 6 \beta_{2} + 18 \beta_1 - 170) q^{73} + ( - 6 \beta_{2} - 14 \beta_1 - 260) q^{77} + ( - 11 \beta_{2} - 9 \beta_1 + 530) q^{79} + (10 \beta_{2} + 40 \beta_1 - 238) q^{83} + (5 \beta_{2} + 25 \beta_1 + 50) q^{85} + (8 \beta_{2} + 16 \beta_1 + 220) q^{89} + (14 \beta_{2} - 46 \beta_1 + 660) q^{91} + (15 \beta_{2} - 5 \beta_1) q^{95} + ( - 18 \beta_{2} - 26 \beta_1 - 462) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 15 q^{5} + 14 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 15 q^{5} + 14 q^{7} + 22 q^{11} + 8 q^{13} + 34 q^{17} - 4 q^{19} + 176 q^{23} + 75 q^{25} - 98 q^{29} + 88 q^{31} + 70 q^{35} + 284 q^{37} + 8 q^{41} + 504 q^{43} + 280 q^{47} - 409 q^{49} + 150 q^{53} + 110 q^{55} + 350 q^{59} + 350 q^{61} + 40 q^{65} + 804 q^{67} + 500 q^{71} - 486 q^{73} - 788 q^{77} + 1592 q^{79} - 684 q^{83} + 170 q^{85} + 668 q^{89} + 1920 q^{91} - 20 q^{95} - 1394 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 17x + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\nu^{2} - 47 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 47 ) / 4 \) 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
4.08359
1.06301
−4.14660
0 0 0 5.00000 0 −10.3344 0 0 0
1.2 0 0 0 5.00000 0 1.74795 0 0 0
1.3 0 0 0 5.00000 0 22.5864 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.bk yes 3
3.b odd 2 1 1440.4.a.bi yes 3
4.b odd 2 1 1440.4.a.bj yes 3
12.b even 2 1 1440.4.a.bh 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1440.4.a.bh 3 12.b even 2 1
1440.4.a.bi yes 3 3.b odd 2 1
1440.4.a.bj yes 3 4.b odd 2 1
1440.4.a.bk yes 3 1.a even 1 1 trivial

Hecke kernels

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

\( T_{7}^{3} - 14T_{7}^{2} - 212T_{7} + 408 \) Copy content Toggle raw display
\( T_{11}^{3} - 22T_{11}^{2} - 1844T_{11} - 10632 \) Copy content Toggle raw display
\( T_{17}^{3} - 34T_{17}^{2} - 6788T_{17} - 96888 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T - 5)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 14 T^{2} + \cdots + 408 \) Copy content Toggle raw display
$11$ \( T^{3} - 22 T^{2} + \cdots - 10632 \) Copy content Toggle raw display
$13$ \( T^{3} - 8 T^{2} + \cdots - 84480 \) Copy content Toggle raw display
$17$ \( T^{3} - 34 T^{2} + \cdots - 96888 \) Copy content Toggle raw display
$19$ \( T^{3} + 4 T^{2} + \cdots + 474368 \) Copy content Toggle raw display
$23$ \( T^{3} - 176 T^{2} + \cdots + 30720 \) Copy content Toggle raw display
$29$ \( T^{3} + 98 T^{2} + \cdots + 2277784 \) Copy content Toggle raw display
$31$ \( T^{3} - 88 T^{2} + \cdots + 1707904 \) Copy content Toggle raw display
$37$ \( T^{3} - 284 T^{2} + \cdots + 15742208 \) Copy content Toggle raw display
$41$ \( T^{3} - 8 T^{2} + \cdots - 9332224 \) Copy content Toggle raw display
$43$ \( T^{3} - 504 T^{2} + \cdots + 39200768 \) Copy content Toggle raw display
$47$ \( T^{3} - 280 T^{2} + \cdots + 3734656 \) Copy content Toggle raw display
$53$ \( T^{3} - 150 T^{2} + \cdots - 55880 \) Copy content Toggle raw display
$59$ \( T^{3} - 350 T^{2} + \cdots + 160926744 \) Copy content Toggle raw display
$61$ \( T^{3} - 350 T^{2} + \cdots + 23932568 \) Copy content Toggle raw display
$67$ \( T^{3} - 804 T^{2} + \cdots - 5531328 \) Copy content Toggle raw display
$71$ \( T^{3} - 500 T^{2} + \cdots + 3957312 \) Copy content Toggle raw display
$73$ \( T^{3} + 486 T^{2} + \cdots - 1079800 \) Copy content Toggle raw display
$79$ \( T^{3} - 1592 T^{2} + \cdots - 76128384 \) Copy content Toggle raw display
$83$ \( T^{3} + 684 T^{2} + \cdots - 220672448 \) Copy content Toggle raw display
$89$ \( T^{3} - 668 T^{2} + \cdots + 4766400 \) Copy content Toggle raw display
$97$ \( T^{3} + 1394 T^{2} + \cdots - 105157160 \) Copy content Toggle raw display
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