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Properties

Label 1040.2.bv.e
Level $1040$
Weight $2$
Character orbit 1040.bv
Analytic conductor $8.304$
Analytic rank $0$
Dimension $32$
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(207,1040)]
 
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, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1040.207");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.bv (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: \(32\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 36 q^{13} + 56 q^{17} - 8 q^{25} + 16 q^{53} + 112 q^{61} + 28 q^{65} - 88 q^{77} - 80 q^{81}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
207.1 0 −2.15055 − 2.15055i 0 −1.75992 − 1.37938i 0 0.818354 + 0.818354i 0 6.24977i 0
207.2 0 −2.15055 − 2.15055i 0 1.75992 + 1.37938i 0 −0.818354 − 0.818354i 0 6.24977i 0
207.3 0 −1.35608 − 1.35608i 0 2.03436 + 0.928104i 0 −1.50017 − 1.50017i 0 0.677918i 0
207.4 0 −1.35608 − 1.35608i 0 −2.03436 − 0.928104i 0 1.50017 + 1.50017i 0 0.677918i 0
207.5 0 −0.688596 − 0.688596i 0 1.49744 − 1.66062i 0 −2.17463 − 2.17463i 0 − 2.05167i 0
207.6 0 −0.688596 − 0.688596i 0 −1.49744 + 1.66062i 0 2.17463 + 2.17463i 0 − 2.05167i 0
207.7 0 −0.248983 − 0.248983i 0 −0.147459 + 2.23120i 0 0.592245 + 0.592245i 0 − 2.87602i 0
207.8 0 −0.248983 − 0.248983i 0 0.147459 − 2.23120i 0 −0.592245 − 0.592245i 0 − 2.87602i 0
207.9 0 0.248983 + 0.248983i 0 0.147459 − 2.23120i 0 0.592245 + 0.592245i 0 − 2.87602i 0
207.10 0 0.248983 + 0.248983i 0 −0.147459 + 2.23120i 0 −0.592245 − 0.592245i 0 − 2.87602i 0
207.11 0 0.688596 + 0.688596i 0 −1.49744 + 1.66062i 0 −2.17463 − 2.17463i 0 − 2.05167i 0
207.12 0 0.688596 + 0.688596i 0 1.49744 − 1.66062i 0 2.17463 + 2.17463i 0 − 2.05167i 0
207.13 0 1.35608 + 1.35608i 0 −2.03436 − 0.928104i 0 −1.50017 − 1.50017i 0 0.677918i 0
207.14 0 1.35608 + 1.35608i 0 2.03436 + 0.928104i 0 1.50017 + 1.50017i 0 0.677918i 0
207.15 0 2.15055 + 2.15055i 0 1.75992 + 1.37938i 0 0.818354 + 0.818354i 0 6.24977i 0
207.16 0 2.15055 + 2.15055i 0 −1.75992 − 1.37938i 0 −0.818354 − 0.818354i 0 6.24977i 0
623.1 0 −2.15055 + 2.15055i 0 −1.75992 + 1.37938i 0 0.818354 − 0.818354i 0 − 6.24977i 0
623.2 0 −2.15055 + 2.15055i 0 1.75992 − 1.37938i 0 −0.818354 + 0.818354i 0 − 6.24977i 0
623.3 0 −1.35608 + 1.35608i 0 2.03436 − 0.928104i 0 −1.50017 + 1.50017i 0 − 0.677918i 0
623.4 0 −1.35608 + 1.35608i 0 −2.03436 + 0.928104i 0 1.50017 − 1.50017i 0 − 0.677918i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 207.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
13.b even 2 1 inner
20.e even 4 1 inner
52.b odd 2 1 inner
65.h odd 4 1 inner
260.p 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.bv.e ✓ 32
4.b odd 2 1 inner 1040.2.bv.e ✓ 32
5.c odd 4 1 inner 1040.2.bv.e ✓ 32
13.b even 2 1 inner 1040.2.bv.e ✓ 32
20.e even 4 1 inner 1040.2.bv.e ✓ 32
52.b odd 2 1 inner 1040.2.bv.e ✓ 32
65.h odd 4 1 inner 1040.2.bv.e ✓ 32
260.p even 4 1 inner 1040.2.bv.e ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1040.2.bv.e ✓ 32 1.a even 1 1 trivial
1040.2.bv.e ✓ 32 4.b odd 2 1 inner
1040.2.bv.e ✓ 32 5.c odd 4 1 inner
1040.2.bv.e ✓ 32 13.b even 2 1 inner
1040.2.bv.e ✓ 32 20.e even 4 1 inner
1040.2.bv.e ✓ 32 52.b odd 2 1 inner
1040.2.bv.e ✓ 32 65.h odd 4 1 inner
1040.2.bv.e ✓ 32 260.p 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 \(S_{2}^{\mathrm{new}}(1040, [\chi])\):

\( T_{3}^{16} + 100T_{3}^{12} + 1248T_{3}^{8} + 1060T_{3}^{4} + 16 \) Copy content Toggle raw display
\( T_{37}^{16} + 7752T_{37}^{12} + 13341024T_{37}^{8} + 5997911760T_{37}^{4} + 119688321600 \) Copy content Toggle raw display