Properties

Label 483.2.a.j
Level $483$
Weight $2$
Character orbit 483.a
Self dual yes
Analytic conductor $3.857$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [483,2,Mod(1,483)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(483, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("483.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.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: \(3.85677441763\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.15317.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 5x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + \beta_1 + 1) q^{4} + ( - \beta_{3} + \beta_{2} + 2) q^{5} - \beta_1 q^{6} - q^{7} + (\beta_{3} + \beta_{2} + \beta_1 + 2) q^{8} + q^{9} + ( - \beta_{2} + 2 \beta_1 - 1) q^{10}+ \cdots + ( - \beta_{3} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 4 q^{3} + 4 q^{4} + 5 q^{5} - 2 q^{6} - 4 q^{7} + 9 q^{8} + 4 q^{9} + 2 q^{10} + q^{11} - 4 q^{12} + 7 q^{13} - 2 q^{14} - 5 q^{15} + 8 q^{16} + 2 q^{17} + 2 q^{18} - q^{19} + 13 q^{20}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta _1 + 2 \) 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
−1.69353
−0.329727
1.32973
2.69353
−1.69353 −1.00000 0.868028 3.51256 1.69353 −1.00000 1.91702 1.00000 −5.94860
1.2 −0.329727 −1.00000 −1.89128 −2.73589 0.329727 −1.00000 1.28306 1.00000 0.902098
1.3 1.32973 −1.00000 −0.231826 3.17434 −1.32973 −1.00000 −2.96772 1.00000 4.22101
1.4 2.69353 −1.00000 5.25508 1.04900 −2.69353 −1.00000 8.76763 1.00000 2.82550
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(7\) \( +1 \)
\(23\) \( -1 \)

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 483.2.a.j 4
3.b odd 2 1 1449.2.a.o 4
4.b odd 2 1 7728.2.a.ce 4
7.b odd 2 1 3381.2.a.x 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.a.j 4 1.a even 1 1 trivial
1449.2.a.o 4 3.b odd 2 1
3381.2.a.x 4 7.b odd 2 1
7728.2.a.ce 4 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(483))\):

\( T_{2}^{4} - 2T_{2}^{3} - 4T_{2}^{2} + 5T_{2} + 2 \) Copy content Toggle raw display
\( T_{5}^{4} - 5T_{5}^{3} - 3T_{5}^{2} + 38T_{5} - 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + \cdots + 2 \) Copy content Toggle raw display
$3$ \( (T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 5 T^{3} + \cdots - 32 \) Copy content Toggle raw display
$7$ \( (T + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - T^{3} - 17 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$13$ \( T^{4} - 7 T^{3} + \cdots - 188 \) Copy content Toggle raw display
$17$ \( T^{4} - 2 T^{3} + \cdots - 64 \) Copy content Toggle raw display
$19$ \( T^{4} + T^{3} - 17 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$23$ \( (T - 1)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - T - 38)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 6 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$37$ \( T^{4} - 16 T^{3} + \cdots - 188 \) Copy content Toggle raw display
$41$ \( T^{4} - 5 T^{3} + \cdots - 134 \) Copy content Toggle raw display
$43$ \( T^{4} - 9 T^{3} + \cdots - 16 \) Copy content Toggle raw display
$47$ \( T^{4} + 21 T^{3} + \cdots - 1696 \) Copy content Toggle raw display
$53$ \( T^{4} - 10 T^{3} + \cdots + 3578 \) Copy content Toggle raw display
$59$ \( T^{4} + 26 T^{3} + \cdots + 1556 \) Copy content Toggle raw display
$61$ \( T^{4} - 2 T^{3} + \cdots + 1114 \) Copy content Toggle raw display
$67$ \( T^{4} - 5 T^{3} + \cdots + 11008 \) Copy content Toggle raw display
$71$ \( T^{4} + 19 T^{3} + \cdots + 32 \) Copy content Toggle raw display
$73$ \( T^{4} - 10 T^{3} + \cdots + 608 \) Copy content Toggle raw display
$79$ \( T^{4} + 6 T^{3} + \cdots + 8992 \) Copy content Toggle raw display
$83$ \( T^{4} + 2 T^{3} + \cdots + 8152 \) Copy content Toggle raw display
$89$ \( T^{4} + 17 T^{3} + \cdots + 11188 \) Copy content Toggle raw display
$97$ \( T^{4} - 32 T^{3} + \cdots + 1684 \) Copy content Toggle raw display
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