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Properties

Label 4020.2.f.a
Level $4020$
Weight $2$
Character orbit 4020.f
Analytic conductor $32.100$
Analytic rank $0$
Dimension $46$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4020,2,Mod(401,4020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4020, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4020.401");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4020.f (of order \(2\), degree \(1\), 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: \(32.0998616126\)
Analytic rank: \(0\)
Dimension: \(46\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 46 q - 46 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 46 q - 46 q^{5} - 4 q^{9} + 8 q^{19} + 46 q^{25} - 18 q^{27} + 4 q^{33} - 8 q^{37} - 12 q^{39} - 4 q^{41} + 4 q^{45} - 62 q^{49} - 8 q^{51} - 8 q^{53} - 12 q^{57} - 10 q^{63} - 14 q^{67} - 40 q^{73} - 12 q^{81} - 4 q^{91} + 2 q^{93} - 8 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1 0 −1.73155 − 0.0417518i 0 −1.00000 0 − 0.963172i 0 2.99651 + 0.144591i 0
401.2 0 −1.73155 + 0.0417518i 0 −1.00000 0 0.963172i 0 2.99651 − 0.144591i 0
401.3 0 −1.72437 − 0.162914i 0 −1.00000 0 4.26819i 0 2.94692 + 0.561849i 0
401.4 0 −1.72437 + 0.162914i 0 −1.00000 0 − 4.26819i 0 2.94692 − 0.561849i 0
401.5 0 −1.65289 − 0.517632i 0 −1.00000 0 0.387683i 0 2.46411 + 1.71118i 0
401.6 0 −1.65289 + 0.517632i 0 −1.00000 0 − 0.387683i 0 2.46411 − 1.71118i 0
401.7 0 −1.45039 − 0.946770i 0 −1.00000 0 − 1.74006i 0 1.20725 + 2.74637i 0
401.8 0 −1.45039 + 0.946770i 0 −1.00000 0 1.74006i 0 1.20725 − 2.74637i 0
401.9 0 −1.40436 − 1.01380i 0 −1.00000 0 − 3.61166i 0 0.944435 + 2.84746i 0
401.10 0 −1.40436 + 1.01380i 0 −1.00000 0 3.61166i 0 0.944435 − 2.84746i 0
401.11 0 −1.30028 − 1.14423i 0 −1.00000 0 3.38238i 0 0.381467 + 2.97565i 0
401.12 0 −1.30028 + 1.14423i 0 −1.00000 0 − 3.38238i 0 0.381467 − 2.97565i 0
401.13 0 −1.15074 − 1.29452i 0 −1.00000 0 2.00748i 0 −0.351574 + 2.97933i 0
401.14 0 −1.15074 + 1.29452i 0 −1.00000 0 − 2.00748i 0 −0.351574 − 2.97933i 0
401.15 0 −0.963300 − 1.43946i 0 −1.00000 0 2.54461i 0 −1.14411 + 2.77327i 0
401.16 0 −0.963300 + 1.43946i 0 −1.00000 0 − 2.54461i 0 −1.14411 − 2.77327i 0
401.17 0 −0.574646 − 1.63395i 0 −1.00000 0 − 4.28520i 0 −2.33956 + 1.87788i 0
401.18 0 −0.574646 + 1.63395i 0 −1.00000 0 4.28520i 0 −2.33956 − 1.87788i 0
401.19 0 −0.370384 − 1.69199i 0 −1.00000 0 0.0825586i 0 −2.72563 + 1.25337i 0
401.20 0 −0.370384 + 1.69199i 0 −1.00000 0 − 0.0825586i 0 −2.72563 − 1.25337i 0
See all 46 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 401.46
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
201.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4020.2.f.a ✓ 46
3.b odd 2 1 4020.2.f.b yes 46
67.b odd 2 1 4020.2.f.b yes 46
201.d even 2 1 inner 4020.2.f.a ✓ 46
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4020.2.f.a ✓ 46 1.a even 1 1 trivial
4020.2.f.a ✓ 46 201.d even 2 1 inner
4020.2.f.b yes 46 3.b odd 2 1
4020.2.f.b yes 46 67.b odd 2 1