↑

Properties

Label 2240.2.bd.a
Level $2240$
Weight $2$
Character orbit 2240.bd
Analytic conductor $17.886$
Analytic rank $0$
Dimension $44$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2240,2,Mod(561,2240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("2240.561");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.bd (of order \(4\), degree \(2\), 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: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 560)
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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q - 12 q^{11} - 8 q^{15} - 8 q^{19} + 24 q^{27} + 12 q^{29} + 28 q^{37} + 44 q^{43} - 44 q^{49} + 8 q^{51} - 12 q^{53} - 24 q^{59} - 16 q^{61} - 28 q^{63} - 40 q^{65} + 28 q^{67} + 40 q^{69} - 12 q^{77} + 16 q^{79} + 20 q^{81} + 16 q^{85} + 88 q^{93} + 32 q^{95} - 28 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
561.1 0 −2.22016 + 2.22016i 0 0.707107 + 0.707107i 0 − 1.00000i 0 − 6.85821i 0
561.2 0 −2.18833 + 2.18833i 0 −0.707107 − 0.707107i 0 − 1.00000i 0 − 6.57756i 0
561.3 0 −1.92081 + 1.92081i 0 0.707107 + 0.707107i 0 − 1.00000i 0 − 4.37899i 0
561.4 0 −1.63220 + 1.63220i 0 −0.707107 − 0.707107i 0 − 1.00000i 0 − 2.32815i 0
561.5 0 −1.28029 + 1.28029i 0 0.707107 + 0.707107i 0 − 1.00000i 0 − 0.278310i 0
561.6 0 −0.989765 + 0.989765i 0 0.707107 + 0.707107i 0 − 1.00000i 0 1.04073i 0
561.7 0 −0.925827 + 0.925827i 0 −0.707107 − 0.707107i 0 − 1.00000i 0 1.28569i 0
561.8 0 −0.693100 + 0.693100i 0 −0.707107 − 0.707107i 0 − 1.00000i 0 2.03922i 0
561.9 0 −0.659301 + 0.659301i 0 0.707107 + 0.707107i 0 − 1.00000i 0 2.13064i 0
561.10 0 −0.448521 + 0.448521i 0 0.707107 + 0.707107i 0 − 1.00000i 0 2.59766i 0
561.11 0 −0.257753 + 0.257753i 0 −0.707107 − 0.707107i 0 − 1.00000i 0 2.86713i 0
561.12 0 0.296675 − 0.296675i 0 −0.707107 − 0.707107i 0 − 1.00000i 0 2.82397i 0
561.13 0 0.576404 − 0.576404i 0 0.707107 + 0.707107i 0 − 1.00000i 0 2.33552i 0
561.14 0 0.605289 − 0.605289i 0 −0.707107 − 0.707107i 0 − 1.00000i 0 2.26725i 0
561.15 0 0.839605 − 0.839605i 0 −0.707107 − 0.707107i 0 − 1.00000i 0 1.59013i 0
561.16 0 1.16279 − 1.16279i 0 0.707107 + 0.707107i 0 − 1.00000i 0 0.295857i 0
561.17 0 1.25769 − 1.25769i 0 0.707107 + 0.707107i 0 − 1.00000i 0 − 0.163545i 0
561.18 0 1.35880 − 1.35880i 0 −0.707107 − 0.707107i 0 − 1.00000i 0 − 0.692696i 0
561.19 0 1.42818 − 1.42818i 0 0.707107 + 0.707107i 0 − 1.00000i 0 − 1.07940i 0
561.20 0 1.67958 − 1.67958i 0 0.707107 + 0.707107i 0 − 1.00000i 0 − 2.64195i 0
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 561.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.2.bd.a 44
4.b odd 2 1 560.2.bd.a ✓ 44
16.e even 4 1 inner 2240.2.bd.a 44
16.f odd 4 1 560.2.bd.a ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.bd.a ✓ 44 4.b odd 2 1
560.2.bd.a ✓ 44 16.f odd 4 1
2240.2.bd.a 44 1.a even 1 1 trivial
2240.2.bd.a 44 16.e even 4 1 inner