Properties

Label 4050.2.a.g
Level $4050$
Weight $2$
Character orbit 4050.a
Self dual yes
Analytic conductor $32.339$
Analytic rank $1$
Dimension $1$
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] = [4050,2,Mod(1,4050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4050, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4050.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4050 = 2 \cdot 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4050.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: \(32.3394128186\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 90)
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)
 
\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{7} - q^{8} + 2 q^{11} - 6 q^{13} + q^{14} + q^{16} + 2 q^{17} + 6 q^{19} - 2 q^{22} - q^{23} + 6 q^{26} - q^{28} - 9 q^{29} - 2 q^{31} - q^{32} - 2 q^{34} + 2 q^{37} - 6 q^{38}+ \cdots + 6 q^{98}+O(q^{100}) \) 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
0
−1.00000 0 1.00000 0 0 −1.00000 −1.00000 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 4050.2.a.g 1
3.b odd 2 1 4050.2.a.x 1
5.b even 2 1 4050.2.a.be 1
5.c odd 4 2 810.2.c.c 2
9.c even 3 2 1350.2.e.i 2
9.d odd 6 2 450.2.e.b 2
15.d odd 2 1 4050.2.a.j 1
15.e even 4 2 810.2.c.b 2
45.h odd 6 2 450.2.e.g 2
45.j even 6 2 1350.2.e.a 2
45.k odd 12 4 270.2.i.a 4
45.l even 12 4 90.2.i.a 4
180.v odd 12 4 720.2.by.a 4
180.x even 12 4 2160.2.by.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.i.a 4 45.l even 12 4
270.2.i.a 4 45.k odd 12 4
450.2.e.b 2 9.d odd 6 2
450.2.e.g 2 45.h odd 6 2
720.2.by.a 4 180.v odd 12 4
810.2.c.b 2 15.e even 4 2
810.2.c.c 2 5.c odd 4 2
1350.2.e.a 2 45.j even 6 2
1350.2.e.i 2 9.c even 3 2
2160.2.by.b 4 180.x even 12 4
4050.2.a.g 1 1.a even 1 1 trivial
4050.2.a.j 1 15.d odd 2 1
4050.2.a.x 1 3.b odd 2 1
4050.2.a.be 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4050))\):

\( T_{7} + 1 \) Copy content Toggle raw display
\( T_{11} - 2 \) Copy content Toggle raw display
\( T_{13} + 6 \) Copy content Toggle raw display
\( T_{17} - 2 \) Copy content Toggle raw display
\( T_{23} + 1 \) Copy content Toggle raw display
\( T_{41} - 11 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 1 \) Copy content Toggle raw display
$11$ \( T - 2 \) Copy content Toggle raw display
$13$ \( T + 6 \) Copy content Toggle raw display
$17$ \( T - 2 \) Copy content Toggle raw display
$19$ \( T - 6 \) Copy content Toggle raw display
$23$ \( T + 1 \) Copy content Toggle raw display
$29$ \( T + 9 \) Copy content Toggle raw display
$31$ \( T + 2 \) Copy content Toggle raw display
$37$ \( T - 2 \) Copy content Toggle raw display
$41$ \( T - 11 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T - 7 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T - 4 \) Copy content Toggle raw display
$61$ \( T + 7 \) Copy content Toggle raw display
$67$ \( T + 11 \) Copy content Toggle raw display
$71$ \( T - 6 \) Copy content Toggle raw display
$73$ \( T + 4 \) Copy content Toggle raw display
$79$ \( T + 12 \) Copy content Toggle raw display
$83$ \( T - 11 \) Copy content Toggle raw display
$89$ \( T + 1 \) Copy content Toggle raw display
$97$ \( T + 8 \) Copy content Toggle raw display
show more
show less