Properties

Label 2008.2.a.d
Level $2008$
Weight $2$
Character orbit 2008.a
Self dual yes
Analytic conductor $16.034$
Analytic rank $0$
Dimension $23$
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] = [2008,2,Mod(1,2008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2008, 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("2008.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2008 = 2^{3} \cdot 251 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2008.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: \(16.0339607259\)
Analytic rank: \(0\)
Dimension: \(23\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9} + 8 q^{11} + 8 q^{13} + 7 q^{15} + 19 q^{17} - 9 q^{19} + 9 q^{21} + 21 q^{23} + 65 q^{25} + 5 q^{27} + 10 q^{29} - 9 q^{31} + 34 q^{33} + 12 q^{35} + 11 q^{37} - 9 q^{39} + 35 q^{41} - 9 q^{43} + 29 q^{45} + 37 q^{47} + 77 q^{49} - 17 q^{51} + 38 q^{53} - 20 q^{55} + 51 q^{57} + 17 q^{59} + 22 q^{63} + 41 q^{65} + 9 q^{67} + 8 q^{69} + 13 q^{71} + 41 q^{73} + 25 q^{75} + 36 q^{77} - 36 q^{79} + 127 q^{81} + 29 q^{83} + 34 q^{85} + 10 q^{87} + 36 q^{89} - 6 q^{91} + 36 q^{93} + 25 q^{95} + 40 q^{97} + 19 q^{99}+O(q^{100}) \) 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 −3.39399 0 −1.84335 0 −1.40053 0 8.51918 0
1.2 0 −3.22331 0 3.75978 0 2.77250 0 7.38975 0
1.3 0 −3.19806 0 −2.92729 0 4.80027 0 7.22760 0
1.4 0 −2.74485 0 2.14962 0 −3.54966 0 4.53421 0
1.5 0 −2.21536 0 3.09704 0 −4.67984 0 1.90784 0
1.6 0 −2.13165 0 0.257423 0 0.578263 0 1.54395 0
1.7 0 −1.28390 0 3.72143 0 −0.978625 0 −1.35160 0
1.8 0 −1.19399 0 −1.76720 0 −4.20476 0 −1.57440 0
1.9 0 −1.15579 0 −4.09592 0 −0.621313 0 −1.66414 0
1.10 0 −0.347951 0 −1.66668 0 3.92094 0 −2.87893 0
1.11 0 −0.201914 0 −1.79819 0 3.34181 0 −2.95923 0
1.12 0 −0.0812979 0 4.08000 0 4.55548 0 −2.99339 0
1.13 0 0.259421 0 −1.96371 0 −4.76268 0 −2.93270 0
1.14 0 0.452441 0 1.91039 0 1.96124 0 −2.79530 0
1.15 0 1.39383 0 −3.54571 0 −3.34862 0 −1.05724 0
1.16 0 1.63778 0 2.87336 0 −2.53418 0 −0.317671 0
1.17 0 1.96174 0 0.807414 0 3.34299 0 0.848432 0
1.18 0 2.51727 0 −0.472191 0 4.16870 0 3.33662 0
1.19 0 2.53334 0 1.50187 0 0.478852 0 3.41783 0
1.20 0 2.67736 0 4.29688 0 −4.01755 0 4.16827 0
See all 23 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.23
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(251\) \(1\)

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 2008.2.a.d 23
4.b odd 2 1 4016.2.a.m 23
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2008.2.a.d 23 1.a even 1 1 trivial
4016.2.a.m 23 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{23} - 2 T_{3}^{22} - 55 T_{3}^{21} + 107 T_{3}^{20} + 1286 T_{3}^{19} - 2405 T_{3}^{18} - 16716 T_{3}^{17} + 29527 T_{3}^{16} + 132795 T_{3}^{15} - 215665 T_{3}^{14} - 668901 T_{3}^{13} + 957238 T_{3}^{12} + \cdots - 1408 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2008))\). Copy content Toggle raw display