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Properties

Label 8020.2.a.e
Level $8020$
Weight $2$
Character orbit 8020.a
Self dual yes
Analytic conductor $64.040$
Analytic rank $0$
Dimension $35$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8020,2,Mod(1,8020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8020, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8020.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8020 = 2^{2} \cdot 5 \cdot 401 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8020.a (trivial)

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: yes
Analytic conductor: \(64.0400224211\)
Analytic rank: \(0\)
Dimension: \(35\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9} - 2 q^{11} - q^{13} + q^{15} + 18 q^{17} + 2 q^{19} + 12 q^{21} + 13 q^{23} + 35 q^{25} - 7 q^{27} + 25 q^{29} + 13 q^{31} + 14 q^{33} - 6 q^{35} - 19 q^{37} - 3 q^{39} + 24 q^{41} - 5 q^{43} - 52 q^{45} + 19 q^{47} + 55 q^{49} + 41 q^{53} + 2 q^{55} + 14 q^{57} + 3 q^{59} + 13 q^{61} + 70 q^{63} + q^{65} - 17 q^{67} + 64 q^{69} + 17 q^{71} - 63 q^{73} - q^{75} + 54 q^{77} + 11 q^{79} + 107 q^{81} - 8 q^{83} - 18 q^{85} + 36 q^{87} + 38 q^{89} - 27 q^{91} + q^{93} - 2 q^{95} - 54 q^{97} - 51 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 0 −3.41155 0 −1.00000 0 0.311908 0 8.63864 0
1.2 0 −3.20562 0 −1.00000 0 −2.96453 0 7.27598 0
1.3 0 −3.08956 0 −1.00000 0 4.75064 0 6.54541 0
1.4 0 −2.93370 0 −1.00000 0 4.42731 0 5.60662 0
1.5 0 −2.91827 0 −1.00000 0 −4.31238 0 5.51629 0
1.6 0 −2.89929 0 −1.00000 0 2.68176 0 5.40586 0
1.7 0 −2.36832 0 −1.00000 0 −4.12967 0 2.60896 0
1.8 0 −2.25358 0 −1.00000 0 −0.859031 0 2.07862 0
1.9 0 −1.64682 0 −1.00000 0 1.19631 0 −0.287991 0
1.10 0 −1.51742 0 −1.00000 0 −1.44284 0 −0.697439 0
1.11 0 −1.37216 0 −1.00000 0 −1.82698 0 −1.11719 0
1.12 0 −1.36773 0 −1.00000 0 2.86190 0 −1.12932 0
1.13 0 −1.28894 0 −1.00000 0 0.0476003 0 −1.33863 0
1.14 0 −0.839936 0 −1.00000 0 3.45440 0 −2.29451 0
1.15 0 −0.713769 0 −1.00000 0 −1.26919 0 −2.49053 0
1.16 0 −0.505617 0 −1.00000 0 −0.615205 0 −2.74435 0
1.17 0 −0.478839 0 −1.00000 0 1.59155 0 −2.77071 0
1.18 0 −0.167962 0 −1.00000 0 2.31084 0 −2.97179 0
1.19 0 0.0559400 0 −1.00000 0 −0.736794 0 −2.99687 0
1.20 0 0.309116 0 −1.00000 0 −4.23374 0 −2.90445 0
See all 35 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.35
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(401\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8020.2.a.e ✓ 35
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8020.2.a.e ✓ 35 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{35} + T_{3}^{34} - 78 T_{3}^{33} - 74 T_{3}^{32} + 2745 T_{3}^{31} + 2470 T_{3}^{30} + \cdots - 168832 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8020))\). Copy content Toggle raw display