Properties

Label 6008.2.a.e
Level $6008$
Weight $2$
Character orbit 6008.a
Self dual yes
Analytic conductor $47.974$
Analytic rank $0$
Dimension $50$
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] = [6008,2,Mod(1,6008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6008, 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("6008.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6008 = 2^{3} \cdot 751 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6008.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: \(47.9741215344\)
Analytic rank: \(0\)
Dimension: \(50\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) \) 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 −3.22705 0 2.92731 0 0.525757 0 7.41384 0
1.2 0 −2.97826 0 −0.794149 0 −3.63896 0 5.87005 0
1.3 0 −2.97134 0 0.343630 0 −1.43593 0 5.82889 0
1.4 0 −2.96484 0 2.30894 0 −3.32968 0 5.79029 0
1.5 0 −2.83142 0 4.26484 0 3.89972 0 5.01693 0
1.6 0 −2.58697 0 0.0890087 0 4.21791 0 3.69244 0
1.7 0 −2.56656 0 −2.44840 0 −1.80922 0 3.58725 0
1.8 0 −2.49253 0 −3.92718 0 2.18622 0 3.21269 0
1.9 0 −2.41806 0 0.877422 0 2.89114 0 2.84699 0
1.10 0 −2.26403 0 −1.66165 0 −0.979480 0 2.12585 0
1.11 0 −2.24563 0 3.86835 0 −4.12735 0 2.04285 0
1.12 0 −1.56448 0 2.55356 0 −2.27627 0 −0.552416 0
1.13 0 −1.42091 0 −3.43075 0 0.534513 0 −0.981020 0
1.14 0 −1.38456 0 −1.32143 0 2.39545 0 −1.08299 0
1.15 0 −1.34723 0 0.0749039 0 −4.25247 0 −1.18497 0
1.16 0 −1.33859 0 3.94917 0 0.220759 0 −1.20816 0
1.17 0 −1.03653 0 2.53205 0 −0.333645 0 −1.92561 0
1.18 0 −0.691575 0 −1.88995 0 4.98805 0 −2.52172 0
1.19 0 −0.685000 0 2.14563 0 4.94928 0 −2.53078 0
1.20 0 −0.684026 0 −2.03205 0 −0.0739397 0 −2.53211 0
See all 50 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.50
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(751\) \(1\)

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 6008.2.a.e 50
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6008.2.a.e 50 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{50} - 6 T_{3}^{49} - 85 T_{3}^{48} + 562 T_{3}^{47} + 3255 T_{3}^{46} - 24398 T_{3}^{45} - 73438 T_{3}^{44} + 651808 T_{3}^{43} + 1060681 T_{3}^{42} - 12000134 T_{3}^{41} - 9661997 T_{3}^{40} + 161581109 T_{3}^{39} + \cdots - 769024 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6008))\). Copy content Toggle raw display