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Properties

Label 1040.2.cr.d
Level $1040$
Weight $2$
Character orbit 1040.cr
Analytic conductor $8.304$
Analytic rank $0$
Dimension $56$
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: \(56\)
Relative dimension: \(28\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 4 q^{5} + 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q - 4 q^{5} + 56 q^{9} - 8 q^{21} - 32 q^{29} + 16 q^{41} - 60 q^{45} + 48 q^{61} - 68 q^{65} + 104 q^{81} + 16 q^{85} - 88 q^{89}+O(q^{100}) \) 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 −3.08789 0 −2.22534 + 0.218786i 0 −2.09720 − 2.09720i 0 6.53504 0
239.2 0 −3.08789 0 −0.218786 + 2.22534i 0 −2.09720 − 2.09720i 0 6.53504 0
239.3 0 −2.77441 0 −2.23256 − 0.125247i 0 3.54413 + 3.54413i 0 4.69738 0
239.4 0 −2.77441 0 0.125247 + 2.23256i 0 3.54413 + 3.54413i 0 4.69738 0
239.5 0 −2.61002 0 0.864970 − 2.06200i 0 −1.26098 − 1.26098i 0 3.81221 0
239.6 0 −2.61002 0 2.06200 − 0.864970i 0 −1.26098 − 1.26098i 0 3.81221 0
239.7 0 −1.40194 0 −1.05025 − 1.97407i 0 −0.517601 − 0.517601i 0 −1.03457 0
239.8 0 −1.40194 0 1.97407 + 1.05025i 0 −0.517601 − 0.517601i 0 −1.03457 0
239.9 0 −0.951045 0 −0.607524 + 2.15196i 0 −0.0321617 − 0.0321617i 0 −2.09551 0
239.10 0 −0.951045 0 −2.15196 + 0.607524i 0 −0.0321617 − 0.0321617i 0 −2.09551 0
239.11 0 −0.907176 0 2.08560 − 0.806406i 0 2.93240 + 2.93240i 0 −2.17703 0
239.12 0 −0.907176 0 0.806406 − 2.08560i 0 2.93240 + 2.93240i 0 −2.17703 0
239.13 0 −0.512333 0 1.35039 + 1.78226i 0 −1.89285 − 1.89285i 0 −2.73751 0
239.14 0 −0.512333 0 −1.78226 − 1.35039i 0 −1.89285 − 1.89285i 0 −2.73751 0
239.15 0 0.512333 0 −1.78226 − 1.35039i 0 1.89285 + 1.89285i 0 −2.73751 0
239.16 0 0.512333 0 1.35039 + 1.78226i 0 1.89285 + 1.89285i 0 −2.73751 0
239.17 0 0.907176 0 0.806406 − 2.08560i 0 −2.93240 − 2.93240i 0 −2.17703 0
239.18 0 0.907176 0 2.08560 − 0.806406i 0 −2.93240 − 2.93240i 0 −2.17703 0
239.19 0 0.951045 0 −2.15196 + 0.607524i 0 0.0321617 + 0.0321617i 0 −2.09551 0
239.20 0 0.951045 0 −0.607524 + 2.15196i 0 0.0321617 + 0.0321617i 0 −2.09551 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 239.28
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.d ✓ 56
4.b odd 2 1 inner 1040.2.cr.d ✓ 56
5.b even 2 1 inner 1040.2.cr.d ✓ 56
13.d odd 4 1 inner 1040.2.cr.d ✓ 56
20.d odd 2 1 inner 1040.2.cr.d ✓ 56
52.f even 4 1 inner 1040.2.cr.d ✓ 56
65.g odd 4 1 inner 1040.2.cr.d ✓ 56
260.u even 4 1 inner 1040.2.cr.d ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1040.2.cr.d ✓ 56 1.a even 1 1 trivial
1040.2.cr.d ✓ 56 4.b odd 2 1 inner
1040.2.cr.d ✓ 56 5.b even 2 1 inner
1040.2.cr.d ✓ 56 13.d odd 4 1 inner
1040.2.cr.d ✓ 56 20.d odd 2 1 inner
1040.2.cr.d ✓ 56 52.f even 4 1 inner
1040.2.cr.d ✓ 56 65.g odd 4 1 inner
1040.2.cr.d ✓ 56 260.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 \(S_{2}^{\mathrm{new}}(1040, [\chi])\):

\( T_{3}^{14} - 28T_{3}^{12} + 291T_{3}^{10} - 1380T_{3}^{8} + 3014T_{3}^{6} - 3050T_{3}^{4} + 1348T_{3}^{2} - 192 \) Copy content Toggle raw display
\( T_{17}^{14} - 100 T_{17}^{12} + 3553 T_{17}^{10} - 58814 T_{17}^{8} + 465908 T_{17}^{6} - 1487160 T_{17}^{4} + \cdots - 30976 \) Copy content Toggle raw display