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Properties

Label 4140.2.s.a
Level $4140$
Weight $2$
Character orbit 4140.s
Analytic conductor $33.058$
Analytic rank $0$
Dimension $44$
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] = [4140,2,Mod(737,4140)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4140, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4140.737");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4140.s (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: \(33.0580664368\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q - 12 q^{7} - 4 q^{13} + 24 q^{25} - 48 q^{37} + 8 q^{43} + 40 q^{55} - 96 q^{61} - 44 q^{67} + 76 q^{73} + 72 q^{85} - 48 q^{91} - 20 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
737.1 0 0 0 −2.23603 + 0.0138422i 0 2.36435 + 2.36435i 0 0 0
737.2 0 0 0 −2.21141 + 0.331134i 0 −1.17089 − 1.17089i 0 0 0
737.3 0 0 0 −2.20494 + 0.371802i 0 −3.41792 − 3.41792i 0 0 0
737.4 0 0 0 −2.19878 − 0.406643i 0 1.26371 + 1.26371i 0 0 0
737.5 0 0 0 −1.99005 − 1.01966i 0 −1.87601 − 1.87601i 0 0 0
737.6 0 0 0 −1.77166 + 1.36427i 0 3.33197 + 3.33197i 0 0 0
737.7 0 0 0 −1.54074 − 1.62053i 0 −0.122636 − 0.122636i 0 0 0
737.8 0 0 0 −1.13155 + 1.92862i 0 2.43117 + 2.43117i 0 0 0
737.9 0 0 0 −0.340177 − 2.21004i 0 −3.49741 − 3.49741i 0 0 0
737.10 0 0 0 −0.210429 + 2.22614i 0 −0.752416 − 0.752416i 0 0 0
737.11 0 0 0 −0.00942680 + 2.23605i 0 −1.55392 − 1.55392i 0 0 0
737.12 0 0 0 0.00942680 − 2.23605i 0 −1.55392 − 1.55392i 0 0 0
737.13 0 0 0 0.210429 − 2.22614i 0 −0.752416 − 0.752416i 0 0 0
737.14 0 0 0 0.340177 + 2.21004i 0 −3.49741 − 3.49741i 0 0 0
737.15 0 0 0 1.13155 − 1.92862i 0 2.43117 + 2.43117i 0 0 0
737.16 0 0 0 1.54074 + 1.62053i 0 −0.122636 − 0.122636i 0 0 0
737.17 0 0 0 1.77166 − 1.36427i 0 3.33197 + 3.33197i 0 0 0
737.18 0 0 0 1.99005 + 1.01966i 0 −1.87601 − 1.87601i 0 0 0
737.19 0 0 0 2.19878 + 0.406643i 0 1.26371 + 1.26371i 0 0 0
737.20 0 0 0 2.20494 − 0.371802i 0 −3.41792 − 3.41792i 0 0 0
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 737.22
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 4140.2.s.a ✓ 44
3.b odd 2 1 inner 4140.2.s.a ✓ 44
5.c odd 4 1 inner 4140.2.s.a ✓ 44
15.e even 4 1 inner 4140.2.s.a ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4140.2.s.a ✓ 44 1.a even 1 1 trivial
4140.2.s.a ✓ 44 3.b odd 2 1 inner
4140.2.s.a ✓ 44 5.c odd 4 1 inner
4140.2.s.a ✓ 44 15.e even 4 1 inner