↑

Properties

Label 6008.2.a.d
Level $6008$
Weight $2$
Character orbit 6008.a
Self dual yes
Analytic conductor $47.974$
Analytic rank $0$
Dimension $49$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(49\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9} + 19 q^{11} + 15 q^{13} + 17 q^{15} + 14 q^{17} + 24 q^{19} - 8 q^{21} + 28 q^{23} + 72 q^{25} + 62 q^{27} - 35 q^{29} + 51 q^{31} + 28 q^{33} + 23 q^{35} + 19 q^{37} + 34 q^{39} + 12 q^{41} + 37 q^{43} - 20 q^{45} + 54 q^{47} + 65 q^{49} + 43 q^{51} - 17 q^{53} + 57 q^{55} + 19 q^{57} + 52 q^{59} - 16 q^{61} + 41 q^{63} + 13 q^{65} + 44 q^{67} - 4 q^{69} + 52 q^{71} + 58 q^{73} + 81 q^{75} - 27 q^{77} + 43 q^{79} + 73 q^{81} + 51 q^{83} - 16 q^{85} + 41 q^{87} + 40 q^{89} + 73 q^{91} + 22 q^{93} + 70 q^{95} + 96 q^{97} + 78 q^{99}+O(q^{100}) \) 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 −3.18877 0 2.23846 0 2.57450 0 7.16824 0
1.2 0 −2.93504 0 −2.06805 0 2.42601 0 5.61445 0
1.3 0 −2.90482 0 −3.52511 0 3.26846 0 5.43799 0
1.4 0 −2.65403 0 −0.749159 0 −1.02284 0 4.04389 0
1.5 0 −2.55321 0 −4.16598 0 −2.08978 0 3.51888 0
1.6 0 −2.54750 0 −2.18138 0 −2.46793 0 3.48974 0
1.7 0 −2.35470 0 0.425735 0 −0.578482 0 2.54461 0
1.8 0 −2.20855 0 1.60744 0 2.65561 0 1.87768 0
1.9 0 −2.08107 0 −0.324432 0 −2.64730 0 1.33086 0
1.10 0 −1.95337 0 3.19909 0 4.95256 0 0.815657 0
1.11 0 −1.83267 0 0.130732 0 −0.665399 0 0.358681 0
1.12 0 −1.52312 0 3.48921 0 −1.44076 0 −0.680107 0
1.13 0 −1.49420 0 −0.747906 0 4.82128 0 −0.767369 0
1.14 0 −1.42908 0 1.27830 0 −2.61604 0 −0.957738 0
1.15 0 −1.11974 0 2.58610 0 0.902449 0 −1.74619 0
1.16 0 −1.08636 0 −0.0864029 0 −0.643099 0 −1.81982 0
1.17 0 −1.04405 0 −3.83943 0 4.18991 0 −1.90997 0
1.18 0 −1.04212 0 −3.77557 0 1.06595 0 −1.91399 0
1.19 0 −0.502181 0 −0.673128 0 2.13432 0 −2.74781 0
1.20 0 −0.385942 0 −1.81441 0 −4.21344 0 −2.85105 0
See all 49 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.49
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.d ✓ 49
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6008.2.a.d ✓ 49 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{49} - 14 T_{3}^{48} - 5 T_{3}^{47} + 936 T_{3}^{46} - 2879 T_{3}^{45} - 26512 T_{3}^{44} + \cdots - 25459200 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6008))\). Copy content Toggle raw display