Properties

Label 4140.2.a.t
Level $4140$
Weight $2$
Character orbit 4140.a
Self dual yes
Analytic conductor $33.058$
Analytic rank $0$
Dimension $5$
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] = [4140,2,Mod(1,4140)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4140, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4140.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4140.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: \(33.0580664368\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.14345904.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 13x^{3} + 34x^{2} - 11x - 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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 a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{5} + (\beta_{3} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{5} + (\beta_{3} + 1) q^{7} + (\beta_{4} - 1) q^{11} + (\beta_{3} - \beta_{2} + 1) q^{13} + (\beta_{2} - \beta_1 - 1) q^{17} + (\beta_{4} + 1) q^{19} - q^{23} + q^{25} - \beta_{2} q^{29} + ( - 2 \beta_{4} + \beta_{3} - \beta_1 + 2) q^{31} + ( - \beta_{3} - 1) q^{35} + (\beta_{4} + \beta_{2} + 1) q^{37} + (\beta_{4} + \beta_{3} + 2 \beta_1) q^{41} + (\beta_{4} - \beta_{3} - \beta_{2} + \cdots + 4) q^{43}+ \cdots + (\beta_{4} - \beta_{3} + \cdots - 2 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 5 q^{5} + 4 q^{7} - 4 q^{11} + 2 q^{13} - 2 q^{17} + 6 q^{19} - 5 q^{23} + 5 q^{25} - 2 q^{29} + 8 q^{31} - 4 q^{35} + 8 q^{37} - 2 q^{41} + 18 q^{43} + 4 q^{47} + 13 q^{49} + 2 q^{53} + 4 q^{55} - 6 q^{59} + 10 q^{61} - 2 q^{65} + 16 q^{67} - 10 q^{71} + 18 q^{73} + 16 q^{77} + 14 q^{79} - 8 q^{83} + 2 q^{85} + 18 q^{89} + 36 q^{91} - 6 q^{95} + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 13x^{3} + 34x^{2} - 11x - 12 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{4} + 12\nu^{2} - 9\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{4} + \nu^{3} + 13\nu^{2} - 18\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} - 14\nu^{2} + 7\nu + 14 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{4} - \nu^{3} - 27\nu^{2} + 29\nu + 14 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{4} - \beta_{3} - \beta_{2} - 2\beta _1 + 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{4} - 4\beta_{3} + 7\beta_{2} - \beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -21\beta_{4} - 3\beta_{3} - 21\beta_{2} - 28\beta _1 + 140 ) / 2 \) 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
1.84503
3.18817
−3.83694
1.23445
−0.430705
0 0 0 −1.00000 0 −3.57723 0 0 0
1.2 0 0 0 −1.00000 0 −0.334658 0 0 0
1.3 0 0 0 −1.00000 0 −0.113901 0 0 0
1.4 0 0 0 −1.00000 0 2.81459 0 0 0
1.5 0 0 0 −1.00000 0 5.21120 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(5\) \( +1 \)
\(23\) \( +1 \)

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 4140.2.a.t 5
3.b odd 2 1 4140.2.a.u yes 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4140.2.a.t 5 1.a even 1 1 trivial
4140.2.a.u yes 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4140))\):

\( T_{7}^{5} - 4T_{7}^{4} - 16T_{7}^{3} + 46T_{7}^{2} + 23T_{7} + 2 \) Copy content Toggle raw display
\( T_{11}^{5} + 4T_{11}^{4} - 26T_{11}^{3} - 108T_{11}^{2} + 84T_{11} + 432 \) Copy content Toggle raw display
\( T_{13}^{5} - 2T_{13}^{4} - 50T_{13}^{3} + 88T_{13}^{2} + 372T_{13} - 72 \) Copy content Toggle raw display
\( T_{17}^{5} + 2T_{17}^{4} - 44T_{17}^{3} - 156T_{17}^{2} + 119T_{17} + 666 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} \) Copy content Toggle raw display
$5$ \( (T + 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} - 4 T^{4} + \cdots + 2 \) Copy content Toggle raw display
$11$ \( T^{5} + 4 T^{4} + \cdots + 432 \) Copy content Toggle raw display
$13$ \( T^{5} - 2 T^{4} + \cdots - 72 \) Copy content Toggle raw display
$17$ \( T^{5} + 2 T^{4} + \cdots + 666 \) Copy content Toggle raw display
$19$ \( T^{5} - 6 T^{4} + \cdots + 72 \) Copy content Toggle raw display
$23$ \( (T + 1)^{5} \) Copy content Toggle raw display
$29$ \( T^{5} + 2 T^{4} + \cdots + 252 \) Copy content Toggle raw display
$31$ \( T^{5} - 8 T^{4} + \cdots - 112 \) Copy content Toggle raw display
$37$ \( T^{5} - 8 T^{4} + \cdots + 122 \) Copy content Toggle raw display
$41$ \( T^{5} + 2 T^{4} + \cdots - 18048 \) Copy content Toggle raw display
$43$ \( T^{5} - 18 T^{4} + \cdots - 3744 \) Copy content Toggle raw display
$47$ \( T^{5} - 4 T^{4} + \cdots + 7344 \) Copy content Toggle raw display
$53$ \( T^{5} - 2 T^{4} + \cdots - 6426 \) Copy content Toggle raw display
$59$ \( T^{5} + 6 T^{4} + \cdots + 18804 \) Copy content Toggle raw display
$61$ \( T^{5} - 10 T^{4} + \cdots + 376 \) Copy content Toggle raw display
$67$ \( T^{5} - 16 T^{4} + \cdots + 27618 \) Copy content Toggle raw display
$71$ \( T^{5} + 10 T^{4} + \cdots + 816 \) Copy content Toggle raw display
$73$ \( T^{5} - 18 T^{4} + \cdots - 136 \) Copy content Toggle raw display
$79$ \( T^{5} - 14 T^{4} + \cdots - 4608 \) Copy content Toggle raw display
$83$ \( T^{5} + 8 T^{4} + \cdots + 33876 \) Copy content Toggle raw display
$89$ \( T^{5} - 18 T^{4} + \cdots + 1152 \) Copy content Toggle raw display
$97$ \( T^{5} - 6 T^{4} + \cdots - 32 \) Copy content Toggle raw display
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