Properties

Label 6240.2.w.g
Level $6240$
Weight $2$
Character orbit 6240.w
Analytic conductor $49.827$
Analytic rank $0$
Dimension $22$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6240,2,Mod(3121,6240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("6240.3121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6240.w (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(49.8266508613\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: no (minimal twist has level 1560)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 4 q^{7} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q + 4 q^{7} - 22 q^{9} + 22 q^{15} + 4 q^{17} - 28 q^{23} - 22 q^{25} - 12 q^{31} + 22 q^{39} + 36 q^{47} + 6 q^{49} - 12 q^{57} - 4 q^{63} - 22 q^{65} - 20 q^{71} - 32 q^{73} + 36 q^{79} + 22 q^{81} - 36 q^{87} + 12 q^{95} + 40 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3121.1 0 1.00000i 0 1.00000i 0 0.423155 0 −1.00000 0
3121.2 0 1.00000i 0 1.00000i 0 0.423155 0 −1.00000 0
3121.3 0 1.00000i 0 1.00000i 0 −3.15960 0 −1.00000 0
3121.4 0 1.00000i 0 1.00000i 0 −3.15960 0 −1.00000 0
3121.5 0 1.00000i 0 1.00000i 0 −2.21683 0 −1.00000 0
3121.6 0 1.00000i 0 1.00000i 0 −2.21683 0 −1.00000 0
3121.7 0 1.00000i 0 1.00000i 0 2.69342 0 −1.00000 0
3121.8 0 1.00000i 0 1.00000i 0 2.69342 0 −1.00000 0
3121.9 0 1.00000i 0 1.00000i 0 1.44049 0 −1.00000 0
3121.10 0 1.00000i 0 1.00000i 0 1.44049 0 −1.00000 0
3121.11 0 1.00000i 0 1.00000i 0 1.58633 0 −1.00000 0
3121.12 0 1.00000i 0 1.00000i 0 1.58633 0 −1.00000 0
3121.13 0 1.00000i 0 1.00000i 0 0.389914 0 −1.00000 0
3121.14 0 1.00000i 0 1.00000i 0 0.389914 0 −1.00000 0
3121.15 0 1.00000i 0 1.00000i 0 −2.83798 0 −1.00000 0
3121.16 0 1.00000i 0 1.00000i 0 −2.83798 0 −1.00000 0
3121.17 0 1.00000i 0 1.00000i 0 −3.83932 0 −1.00000 0
3121.18 0 1.00000i 0 1.00000i 0 −3.83932 0 −1.00000 0
3121.19 0 1.00000i 0 1.00000i 0 2.79776 0 −1.00000 0
3121.20 0 1.00000i 0 1.00000i 0 2.79776 0 −1.00000 0
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3121.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6240.2.w.g 22
4.b odd 2 1 1560.2.w.g 22
8.b even 2 1 inner 6240.2.w.g 22
8.d odd 2 1 1560.2.w.g 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.w.g 22 4.b odd 2 1
1560.2.w.g 22 8.d odd 2 1
6240.2.w.g 22 1.a even 1 1 trivial
6240.2.w.g 22 8.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(6240, [\chi])\):

\( T_{7}^{11} - 2 T_{7}^{10} - 38 T_{7}^{9} + 68 T_{7}^{8} + 485 T_{7}^{7} - 886 T_{7}^{6} - 2396 T_{7}^{5} + 5052 T_{7}^{4} + 2740 T_{7}^{3} - 10208 T_{7}^{2} + 5952 T_{7} - 1024 \) Copy content Toggle raw display
\( T_{11}^{22} + 104 T_{11}^{20} + 4162 T_{11}^{18} + 82844 T_{11}^{16} + 881233 T_{11}^{14} + 4962468 T_{11}^{12} + 13748308 T_{11}^{10} + 16805040 T_{11}^{8} + 9603264 T_{11}^{6} + 2643200 T_{11}^{4} + \cdots + 16384 \) Copy content Toggle raw display