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Properties

Label 1040.2.cr.c
Level $1040$
Weight $2$
Character orbit 1040.cr
Analytic conductor $8.304$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1040,2,Mod(239,1040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1040, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1040.239");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.cr (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: \(8.30444181021\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{5} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{5} + 40 q^{9} + 8 q^{21} + 16 q^{29} + 16 q^{41} + 48 q^{45} + 112 q^{61} + 64 q^{65} - 56 q^{81} - 8 q^{85} + 40 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1 0 −2.97983 0 2.18490 + 0.475623i 0 0.939982 + 0.939982i 0 5.87936 0
239.2 0 −2.97983 0 −0.475623 − 2.18490i 0 0.939982 + 0.939982i 0 5.87936 0
239.3 0 −1.64704 0 1.12588 + 1.93194i 0 −3.42282 − 3.42282i 0 −0.287258 0
239.4 0 −1.64704 0 −1.93194 − 1.12588i 0 −3.42282 − 3.42282i 0 −0.287258 0
239.5 0 −1.55174 0 −2.07838 + 0.824836i 0 1.18353 + 1.18353i 0 −0.592104 0
239.6 0 −1.55174 0 −0.824836 + 2.07838i 0 1.18353 + 1.18353i 0 −0.592104 0
239.7 0 1.55174 0 −0.824836 + 2.07838i 0 −1.18353 − 1.18353i 0 −0.592104 0
239.8 0 1.55174 0 −2.07838 + 0.824836i 0 −1.18353 − 1.18353i 0 −0.592104 0
239.9 0 1.64704 0 −1.93194 − 1.12588i 0 3.42282 + 3.42282i 0 −0.287258 0
239.10 0 1.64704 0 1.12588 + 1.93194i 0 3.42282 + 3.42282i 0 −0.287258 0
239.11 0 2.97983 0 −0.475623 − 2.18490i 0 −0.939982 − 0.939982i 0 5.87936 0
239.12 0 2.97983 0 2.18490 + 0.475623i 0 −0.939982 − 0.939982i 0 5.87936 0
879.1 0 −2.97983 0 2.18490 − 0.475623i 0 0.939982 − 0.939982i 0 5.87936 0
879.2 0 −2.97983 0 −0.475623 + 2.18490i 0 0.939982 − 0.939982i 0 5.87936 0
879.3 0 −1.64704 0 1.12588 − 1.93194i 0 −3.42282 + 3.42282i 0 −0.287258 0
879.4 0 −1.64704 0 −1.93194 + 1.12588i 0 −3.42282 + 3.42282i 0 −0.287258 0
879.5 0 −1.55174 0 −2.07838 − 0.824836i 0 1.18353 − 1.18353i 0 −0.592104 0
879.6 0 −1.55174 0 −0.824836 − 2.07838i 0 1.18353 − 1.18353i 0 −0.592104 0
879.7 0 1.55174 0 −0.824836 − 2.07838i 0 −1.18353 + 1.18353i 0 −0.592104 0
879.8 0 1.55174 0 −2.07838 − 0.824836i 0 −1.18353 + 1.18353i 0 −0.592104 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 239.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
13.d odd 4 1 inner
20.d odd 2 1 inner
52.f even 4 1 inner
65.g odd 4 1 inner
260.u even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1040.2.cr.c ✓ 24
4.b odd 2 1 inner 1040.2.cr.c ✓ 24
5.b even 2 1 inner 1040.2.cr.c ✓ 24
13.d odd 4 1 inner 1040.2.cr.c ✓ 24
20.d odd 2 1 inner 1040.2.cr.c ✓ 24
52.f even 4 1 inner 1040.2.cr.c ✓ 24
65.g odd 4 1 inner 1040.2.cr.c ✓ 24
260.u even 4 1 inner 1040.2.cr.c ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1040.2.cr.c ✓ 24 1.a even 1 1 trivial
1040.2.cr.c ✓ 24 4.b odd 2 1 inner
1040.2.cr.c ✓ 24 5.b even 2 1 inner
1040.2.cr.c ✓ 24 13.d odd 4 1 inner
1040.2.cr.c ✓ 24 20.d odd 2 1 inner
1040.2.cr.c ✓ 24 52.f even 4 1 inner
1040.2.cr.c ✓ 24 65.g odd 4 1 inner
1040.2.cr.c ✓ 24 260.u even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1040, [\chi])\):

\( T_{3}^{6} - 14T_{3}^{4} + 52T_{3}^{2} - 58 \) Copy content Toggle raw display
\( T_{17}^{6} - 54T_{17}^{4} + 692T_{17}^{2} - 104 \) Copy content Toggle raw display