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: | \(1\) |
Dimension: | \(44\) |
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.
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 | −3.30487 | 0 | 0.148423 | 0 | −4.48376 | 0 | 7.92214 | 0 | ||||||||||||||||||
1.2 | 0 | −3.25541 | 0 | 3.98380 | 0 | −3.42108 | 0 | 7.59768 | 0 | ||||||||||||||||||
1.3 | 0 | −3.25538 | 0 | −1.89943 | 0 | −1.94419 | 0 | 7.59748 | 0 | ||||||||||||||||||
1.4 | 0 | −3.18197 | 0 | −2.41204 | 0 | 2.14400 | 0 | 7.12495 | 0 | ||||||||||||||||||
1.5 | 0 | −2.99842 | 0 | 0.770002 | 0 | 1.05287 | 0 | 5.99050 | 0 | ||||||||||||||||||
1.6 | 0 | −2.98092 | 0 | 3.05916 | 0 | 0.297244 | 0 | 5.88586 | 0 | ||||||||||||||||||
1.7 | 0 | −2.46379 | 0 | 1.54615 | 0 | −4.51355 | 0 | 3.07028 | 0 | ||||||||||||||||||
1.8 | 0 | −2.32283 | 0 | 3.89591 | 0 | 0.0996640 | 0 | 2.39554 | 0 | ||||||||||||||||||
1.9 | 0 | −2.25093 | 0 | 3.53241 | 0 | 1.19008 | 0 | 2.06668 | 0 | ||||||||||||||||||
1.10 | 0 | −2.12121 | 0 | −1.44700 | 0 | 4.49803 | 0 | 1.49952 | 0 | ||||||||||||||||||
1.11 | 0 | −2.09991 | 0 | −2.89043 | 0 | −1.97185 | 0 | 1.40961 | 0 | ||||||||||||||||||
1.12 | 0 | −2.01203 | 0 | −4.06513 | 0 | −4.33062 | 0 | 1.04828 | 0 | ||||||||||||||||||
1.13 | 0 | −2.00709 | 0 | 0.0492234 | 0 | 3.84191 | 0 | 1.02840 | 0 | ||||||||||||||||||
1.14 | 0 | −1.75295 | 0 | 1.89627 | 0 | −3.72975 | 0 | 0.0728254 | 0 | ||||||||||||||||||
1.15 | 0 | −1.61777 | 0 | −2.18000 | 0 | −2.82839 | 0 | −0.382806 | 0 | ||||||||||||||||||
1.16 | 0 | −1.52846 | 0 | 0.276340 | 0 | 2.37933 | 0 | −0.663805 | 0 | ||||||||||||||||||
1.17 | 0 | −1.09930 | 0 | −2.74603 | 0 | 2.16897 | 0 | −1.79154 | 0 | ||||||||||||||||||
1.18 | 0 | −1.03287 | 0 | −2.90339 | 0 | −0.294834 | 0 | −1.93318 | 0 | ||||||||||||||||||
1.19 | 0 | −0.962699 | 0 | 2.73078 | 0 | −4.42659 | 0 | −2.07321 | 0 | ||||||||||||||||||
1.20 | 0 | −0.753945 | 0 | −1.16579 | 0 | 1.19034 | 0 | −2.43157 | 0 | ||||||||||||||||||
See all 44 embeddings |
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.b | ✓ | 44 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6008.2.a.b | ✓ | 44 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{44} + 14 T_{3}^{43} + 13 T_{3}^{42} - 688 T_{3}^{41} - 2782 T_{3}^{40} + 12974 T_{3}^{39} + \cdots + 42752 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6008))\).