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Properties

Label 8044.2.a.a
Level $8044$
Weight $2$
Character orbit 8044.a
Self dual yes
Analytic conductor $64.232$
Analytic rank $1$
Dimension $80$
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] = [8044,2,Mod(1,8044)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8044, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8044.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8044 = 2^{2} \cdot 2011 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8044.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: \(64.2316633859\)
Analytic rank: \(1\)
Dimension: \(80\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9} - 34 q^{11} - q^{13} - 24 q^{15} - 35 q^{17} - 31 q^{19} - 3 q^{21} - 43 q^{23} + 58 q^{25} - 49 q^{27} - 5 q^{29} - 56 q^{31} - 23 q^{33} - 72 q^{35} - 11 q^{37} - 74 q^{39} - 81 q^{41} - 34 q^{43} - 14 q^{45} - 64 q^{47} + 40 q^{49} - 59 q^{51} + 3 q^{53} - 53 q^{55} - 34 q^{57} - 116 q^{59} - 13 q^{61} - 61 q^{63} - 55 q^{65} - 22 q^{67} - 10 q^{69} - 86 q^{71} - 32 q^{73} - 85 q^{75} + 4 q^{77} - 88 q^{79} + 12 q^{81} - 83 q^{83} - 2 q^{85} - 87 q^{87} - 72 q^{89} - 49 q^{91} - 102 q^{95} - 34 q^{97} - 103 q^{99}+O(q^{100}) \) 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 −3.40275 0 −1.72550 0 −2.88747 0 8.57873 0
1.2 0 −3.30826 0 0.889537 0 −1.31364 0 7.94459 0
1.3 0 −3.15920 0 −3.92220 0 2.93790 0 6.98053 0
1.4 0 −3.15577 0 3.07374 0 −1.51261 0 6.95887 0
1.5 0 −3.13382 0 2.56113 0 −0.308664 0 6.82084 0
1.6 0 −3.01017 0 −0.450076 0 4.39052 0 6.06114 0
1.7 0 −2.88063 0 4.02746 0 2.18923 0 5.29800 0
1.8 0 −2.78192 0 1.93177 0 −4.05350 0 4.73908 0
1.9 0 −2.69065 0 −1.88521 0 −0.388125 0 4.23960 0
1.10 0 −2.64514 0 −1.03664 0 1.03871 0 3.99678 0
1.11 0 −2.64034 0 1.50045 0 0.628661 0 3.97141 0
1.12 0 −2.57629 0 −1.13453 0 −4.92532 0 3.63726 0
1.13 0 −2.57058 0 3.02835 0 −3.19846 0 3.60789 0
1.14 0 −2.56684 0 −2.59165 0 1.04792 0 3.58865 0
1.15 0 −2.41417 0 −0.367373 0 2.85158 0 2.82820 0
1.16 0 −2.27117 0 −3.46385 0 −0.730522 0 2.15820 0
1.17 0 −2.04584 0 −1.96310 0 −5.24497 0 1.18544 0
1.18 0 −1.94593 0 4.46459 0 −3.96440 0 0.786636 0
1.19 0 −1.93385 0 −1.81561 0 3.06905 0 0.739771 0
1.20 0 −1.85715 0 0.680485 0 2.30245 0 0.448994 0
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.80
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(2011\) \(-1\)

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 8044.2.a.a ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8044.2.a.a ✓ 80 1.a even 1 1 trivial