Properties

Label 2.48.a.b
Level $2$
Weight $48$
Character orbit 2.a
Self dual yes
Analytic conductor $27.982$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2,48,Mod(1,2)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2, base_ring=CyclotomicField(1))
 
chi = DirichletCharacter(H, H._module([]))
 
N = Newforms(chi, 48, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2.1");
 
S:= CuspForms(chi, 48);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 48 \)
Character orbit: \([\chi]\) \(=\) 2.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: \(27.9815325310\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 5897345978580 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{3}\cdot 5 \)
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 \(\beta = 8640\sqrt{23589383914321}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 8388608 q^{2} + ( - 5 \beta + 61144922412) q^{3} + 70368744177664 q^{4} + (460908 \beta + 90\!\cdots\!70) q^{5}+ \cdots + ( - 611449224120 \beta + 21\!\cdots\!57) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 8388608 q^{2} + ( - 5 \beta + 61144922412) q^{3} + 70368744177664 q^{4} + (460908 \beta + 90\!\cdots\!70) q^{5}+ \cdots + ( - 20\!\cdots\!75 \beta + 98\!\cdots\!44) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16777216 q^{2} + 122289844824 q^{3} + 140737488355328 q^{4} + 18\!\cdots\!40 q^{5}+ \cdots + 42\!\cdots\!14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 16777216 q^{2} + 122289844824 q^{3} + 140737488355328 q^{4} + 18\!\cdots\!40 q^{5}+ \cdots + 19\!\cdots\!88 q^{99}+O(q^{100}) \) 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
2.42845e6
−2.42844e6
−8.38861e6 −1.48673e11 7.03687e13 2.84306e16 1.24716e18 5.21750e19 −5.90296e20 −4.48523e21 −2.38493e23
1.2 −8.38861e6 2.70963e11 7.03687e13 −1.02521e16 −2.27300e18 1.08843e20 −5.90296e20 4.68319e22 8.60007e22
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)

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 2.48.a.b 2
4.b odd 2 1 16.48.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.48.a.b 2 1.a even 1 1 trivial
16.48.a.b 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 122289844824T_{3} - 40284750299492923142256 \) acting on \(S_{48}^{\mathrm{new}}(\Gamma_0(2))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 8388608)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 122289844824 T - 40\!\cdots\!56 \) Copy content Toggle raw display
$5$ \( T^{2} + \cdots - 29\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{2} + \cdots + 56\!\cdots\!56 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots - 10\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots - 33\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots - 58\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 81\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 62\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 49\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 82\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 43\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots - 22\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 30\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 22\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 13\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 16\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 30\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 23\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 24\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 35\!\cdots\!84 \) Copy content Toggle raw display
show more
show less