Properties

Label 8020.2.a.d
Level $8020$
Weight $2$
Character orbit 8020.a
Self dual yes
Analytic conductor $64.040$
Analytic rank $1$
Dimension $29$
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: \(1\)
Dimension: \(29\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9} + 2 q^{11} - 23 q^{13} - 3 q^{15} - 30 q^{17} - 6 q^{19} - 16 q^{21} - 21 q^{23} + 29 q^{25} - 15 q^{27} - 35 q^{29} - 7 q^{31} - 36 q^{33} - 8 q^{35} - 31 q^{37} - 11 q^{39} - 24 q^{41} - 17 q^{43} + 10 q^{45} - 17 q^{47} + q^{49} + 8 q^{51} - 57 q^{53} + 2 q^{55} - 46 q^{57} - 9 q^{59} - 27 q^{61} - 34 q^{63} - 23 q^{65} - 21 q^{67} - 28 q^{69} - 19 q^{71} - 81 q^{73} - 3 q^{75} - 66 q^{77} - 17 q^{79} - 39 q^{81} - 30 q^{83} - 30 q^{85} - 20 q^{87} - 38 q^{89} + q^{91} - 75 q^{93} - 6 q^{95} - 48 q^{97} - 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −3.11347 0 1.00000 0 3.18726 0 6.69370 0
1.2 0 −2.75932 0 1.00000 0 −2.48714 0 4.61387 0
1.3 0 −2.73282 0 1.00000 0 1.81499 0 4.46832 0
1.4 0 −2.62437 0 1.00000 0 −4.07774 0 3.88732 0
1.5 0 −2.50668 0 1.00000 0 −0.362320 0 3.28346 0
1.6 0 −2.25625 0 1.00000 0 0.417578 0 2.09065 0
1.7 0 −1.73579 0 1.00000 0 2.72114 0 0.0129592 0
1.8 0 −1.59991 0 1.00000 0 −1.91437 0 −0.440296 0
1.9 0 −1.35123 0 1.00000 0 0.606242 0 −1.17417 0
1.10 0 −1.12115 0 1.00000 0 0.722262 0 −1.74302 0
1.11 0 −0.848679 0 1.00000 0 −3.71403 0 −2.27974 0
1.12 0 −0.697718 0 1.00000 0 −4.99015 0 −2.51319 0
1.13 0 −0.537747 0 1.00000 0 −1.46052 0 −2.71083 0
1.14 0 −0.202645 0 1.00000 0 1.27188 0 −2.95893 0
1.15 0 −0.107530 0 1.00000 0 2.68164 0 −2.98844 0
1.16 0 0.0219398 0 1.00000 0 2.04593 0 −2.99952 0
1.17 0 0.121745 0 1.00000 0 5.25775 0 −2.98518 0
1.18 0 0.358058 0 1.00000 0 1.78556 0 −2.87179 0
1.19 0 0.540664 0 1.00000 0 −1.71282 0 −2.70768 0
1.20 0 1.03867 0 1.00000 0 −0.0217287 0 −1.92117 0
See all 29 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.29
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.d 29
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8020.2.a.d 29 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{29} + 3 T_{3}^{28} - 44 T_{3}^{27} - 130 T_{3}^{26} + 851 T_{3}^{25} + 2470 T_{3}^{24} - 9512 T_{3}^{23} - 27088 T_{3}^{22} + 67874 T_{3}^{21} + 189949 T_{3}^{20} - 321767 T_{3}^{19} - 891226 T_{3}^{18} + 1019291 T_{3}^{17} + \cdots + 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8020))\). Copy content Toggle raw display