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Properties

Label 8020.2.a.f
Level $8020$
Weight $2$
Character orbit 8020.a
Self dual yes
Analytic conductor $64.040$
Analytic rank $0$
Dimension $37$
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: \(37\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9} + 2 q^{11} + 27 q^{13} + 3 q^{15} + 36 q^{17} - 6 q^{19} + 20 q^{21} + 17 q^{23} + 37 q^{25} + 9 q^{27} + 29 q^{29} + 5 q^{31} + 36 q^{33} + 4 q^{35} + 35 q^{37} + 21 q^{39} + 24 q^{41} + 11 q^{43} + 50 q^{45} + 19 q^{47} + 57 q^{49} + 8 q^{51} + 65 q^{53} + 2 q^{55} + 62 q^{57} - 9 q^{59} + 13 q^{61} + 26 q^{63} + 27 q^{65} + 13 q^{67} + 20 q^{69} + 33 q^{71} + 67 q^{73} + 3 q^{75} + 62 q^{77} + 23 q^{79} + 97 q^{81} + 2 q^{83} + 36 q^{85} + 32 q^{87} + 34 q^{89} + q^{91} + 41 q^{93} - 6 q^{95} + 66 q^{97} - 7 q^{99}+O(q^{100}) \) 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 −3.33477 0 1.00000 0 2.42678 0 8.12069 0
1.2 0 −3.32921 0 1.00000 0 −3.52319 0 8.08363 0
1.3 0 −3.05263 0 1.00000 0 2.86058 0 6.31852 0
1.4 0 −2.74673 0 1.00000 0 −2.35693 0 4.54455 0
1.5 0 −2.47047 0 1.00000 0 −0.726513 0 3.10322 0
1.6 0 −2.35696 0 1.00000 0 −1.59035 0 2.55528 0
1.7 0 −2.32976 0 1.00000 0 −1.33564 0 2.42776 0
1.8 0 −2.32562 0 1.00000 0 −4.92659 0 2.40852 0
1.9 0 −2.16998 0 1.00000 0 4.79835 0 1.70880 0
1.10 0 −1.70544 0 1.00000 0 −1.03894 0 −0.0914601 0
1.11 0 −1.65793 0 1.00000 0 4.10503 0 −0.251260 0
1.12 0 −1.55757 0 1.00000 0 3.26992 0 −0.573979 0
1.13 0 −0.881355 0 1.00000 0 3.19359 0 −2.22321 0
1.14 0 −0.869740 0 1.00000 0 −1.85510 0 −2.24355 0
1.15 0 −0.446086 0 1.00000 0 3.25733 0 −2.80101 0
1.16 0 −0.435929 0 1.00000 0 −3.81252 0 −2.80997 0
1.17 0 −0.210422 0 1.00000 0 −4.28610 0 −2.95572 0
1.18 0 −0.161183 0 1.00000 0 −0.359157 0 −2.97402 0
1.19 0 0.332188 0 1.00000 0 0.507569 0 −2.88965 0
1.20 0 0.353843 0 1.00000 0 1.60118 0 −2.87480 0
See all 37 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.37
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.f ✓ 37
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8020.2.a.f ✓ 37 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{37} - 3 T_{3}^{36} - 76 T_{3}^{35} + 228 T_{3}^{34} + 2605 T_{3}^{33} - 7822 T_{3}^{32} + \cdots - 215296 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8020))\). Copy content Toggle raw display