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Properties

Label 8020.2.a.c
Level $8020$
Weight $2$
Character orbit 8020.a
Self dual yes
Analytic conductor $64.040$
Analytic rank $1$
Dimension $28$
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: \(28\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9} + 2 q^{11} + 3 q^{13} - 3 q^{15} - 10 q^{17} - 2 q^{19} - 12 q^{21} - 23 q^{23} + 28 q^{25} + 9 q^{27} - 37 q^{29} - 11 q^{31} + 2 q^{33} + 4 q^{35} - 3 q^{37} - 19 q^{39} - 30 q^{41} + 13 q^{43} - 17 q^{45} - 15 q^{47} + 12 q^{49} - 8 q^{51} - 35 q^{53} - 2 q^{55} - 22 q^{57} - q^{59} - 33 q^{61} - 20 q^{63} - 3 q^{65} + 19 q^{67} - 8 q^{69} - 31 q^{71} + 31 q^{73} + 3 q^{75} - 42 q^{77} - 29 q^{79} - 36 q^{81} + 14 q^{83} + 10 q^{85} - 32 q^{87} - 32 q^{89} - 7 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97} - 39 q^{99}+O(q^{100}) \) 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 −3.10189 0 −1.00000 0 1.97262 0 6.62170 0
1.2 0 −2.61261 0 −1.00000 0 −1.36796 0 3.82575 0
1.3 0 −2.59496 0 −1.00000 0 1.42420 0 3.73381 0
1.4 0 −2.32143 0 −1.00000 0 0.515797 0 2.38904 0
1.5 0 −2.26002 0 −1.00000 0 −3.39221 0 2.10771 0
1.6 0 −1.62538 0 −1.00000 0 4.42157 0 −0.358130 0
1.7 0 −1.58112 0 −1.00000 0 3.33363 0 −0.500055 0
1.8 0 −1.53415 0 −1.00000 0 0.759833 0 −0.646394 0
1.9 0 −1.48695 0 −1.00000 0 −3.53428 0 −0.788974 0
1.10 0 −1.42561 0 −1.00000 0 −3.82644 0 −0.967639 0
1.11 0 −0.750059 0 −1.00000 0 −4.31732 0 −2.43741 0
1.12 0 −0.324068 0 −1.00000 0 −1.35293 0 −2.89498 0
1.13 0 −0.172341 0 −1.00000 0 3.94901 0 −2.97030 0
1.14 0 0.0547853 0 −1.00000 0 3.87235 0 −2.99700 0
1.15 0 0.167528 0 −1.00000 0 −3.89143 0 −2.97193 0
1.16 0 0.356039 0 −1.00000 0 −0.400497 0 −2.87324 0
1.17 0 0.910150 0 −1.00000 0 1.40834 0 −2.17163 0
1.18 0 1.05681 0 −1.00000 0 3.74444 0 −1.88316 0
1.19 0 1.28191 0 −1.00000 0 −0.791593 0 −1.35670 0
1.20 0 1.67955 0 −1.00000 0 −2.58133 0 −0.179108 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.28
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.c ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8020.2.a.c ✓ 28 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{28} - 3 T_{3}^{27} - 46 T_{3}^{26} + 138 T_{3}^{25} + 923 T_{3}^{24} - 2776 T_{3}^{23} + \cdots - 256 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8020))\). Copy content Toggle raw display