Properties

Label 3420.2.x.c
Level $3420$
Weight $2$
Character orbit 3420.x
Analytic conductor $27.309$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3420,2,Mod(1673,3420)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3420, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3420.1673");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3420.x (of order \(4\), degree \(2\), 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: \(27.3088374913\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{13} - 32 q^{25} + 16 q^{31} + 12 q^{37} - 8 q^{43} + 4 q^{55} - 64 q^{61} - 52 q^{73} + 80 q^{85} - 144 q^{91} - 92 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1673.1 0 0 0 −2.20020 0.398923i 0 −2.63796 + 2.63796i 0 0 0
1673.2 0 0 0 −1.98287 1.03354i 0 2.54134 2.54134i 0 0 0
1673.3 0 0 0 −1.04895 1.97477i 0 0.585090 0.585090i 0 0 0
1673.4 0 0 0 −0.925456 + 2.03557i 0 1.92964 1.92964i 0 0 0
1673.5 0 0 0 −0.401341 + 2.19976i 0 −0.0703440 + 0.0703440i 0 0 0
1673.6 0 0 0 −0.330936 + 2.21144i 0 −2.34776 + 2.34776i 0 0 0
1673.7 0 0 0 0.330936 2.21144i 0 −2.34776 + 2.34776i 0 0 0
1673.8 0 0 0 0.401341 2.19976i 0 −0.0703440 + 0.0703440i 0 0 0
1673.9 0 0 0 0.925456 2.03557i 0 1.92964 1.92964i 0 0 0
1673.10 0 0 0 1.04895 + 1.97477i 0 0.585090 0.585090i 0 0 0
1673.11 0 0 0 1.98287 + 1.03354i 0 2.54134 2.54134i 0 0 0
1673.12 0 0 0 2.20020 + 0.398923i 0 −2.63796 + 2.63796i 0 0 0
2357.1 0 0 0 −2.20020 + 0.398923i 0 −2.63796 2.63796i 0 0 0
2357.2 0 0 0 −1.98287 + 1.03354i 0 2.54134 + 2.54134i 0 0 0
2357.3 0 0 0 −1.04895 + 1.97477i 0 0.585090 + 0.585090i 0 0 0
2357.4 0 0 0 −0.925456 2.03557i 0 1.92964 + 1.92964i 0 0 0
2357.5 0 0 0 −0.401341 2.19976i 0 −0.0703440 0.0703440i 0 0 0
2357.6 0 0 0 −0.330936 2.21144i 0 −2.34776 2.34776i 0 0 0
2357.7 0 0 0 0.330936 + 2.21144i 0 −2.34776 2.34776i 0 0 0
2357.8 0 0 0 0.401341 + 2.19976i 0 −0.0703440 0.0703440i 0 0 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1673.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3420.2.x.c 24
3.b odd 2 1 inner 3420.2.x.c 24
5.c odd 4 1 inner 3420.2.x.c 24
15.e even 4 1 inner 3420.2.x.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3420.2.x.c 24 1.a even 1 1 trivial
3420.2.x.c 24 3.b odd 2 1 inner
3420.2.x.c 24 5.c odd 4 1 inner
3420.2.x.c 24 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} - 10 T_{7}^{9} + 269 T_{7}^{8} - 106 T_{7}^{7} + 50 T_{7}^{6} - 1552 T_{7}^{5} + 16440 T_{7}^{4} - 16020 T_{7}^{3} + 7688 T_{7}^{2} + 1240 T_{7} + 100 \) acting on \(S_{2}^{\mathrm{new}}(3420, [\chi])\). Copy content Toggle raw display