Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4006,2,Mod(1,4006)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4006, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4006.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4006 = 2 \cdot 2003 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4006.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: | \(31.9880710497\) |
Analytic rank: | \(1\) |
Dimension: | \(31\) |
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 | 1.00000 | −3.10750 | 1.00000 | −0.905636 | −3.10750 | 3.10017 | 1.00000 | 6.65658 | −0.905636 | ||||||||||||||||||
1.2 | 1.00000 | −2.96975 | 1.00000 | 3.59055 | −2.96975 | −0.0161745 | 1.00000 | 5.81940 | 3.59055 | ||||||||||||||||||
1.3 | 1.00000 | −2.94307 | 1.00000 | −2.92103 | −2.94307 | −1.64254 | 1.00000 | 5.66165 | −2.92103 | ||||||||||||||||||
1.4 | 1.00000 | −2.69988 | 1.00000 | 2.56179 | −2.69988 | −4.70163 | 1.00000 | 4.28934 | 2.56179 | ||||||||||||||||||
1.5 | 1.00000 | −2.68137 | 1.00000 | 0.326909 | −2.68137 | 2.00522 | 1.00000 | 4.18972 | 0.326909 | ||||||||||||||||||
1.6 | 1.00000 | −2.64840 | 1.00000 | −3.78126 | −2.64840 | −4.25052 | 1.00000 | 4.01402 | −3.78126 | ||||||||||||||||||
1.7 | 1.00000 | −2.59151 | 1.00000 | −4.34351 | −2.59151 | −1.56317 | 1.00000 | 3.71590 | −4.34351 | ||||||||||||||||||
1.8 | 1.00000 | −2.43870 | 1.00000 | 1.19348 | −2.43870 | 2.99932 | 1.00000 | 2.94725 | 1.19348 | ||||||||||||||||||
1.9 | 1.00000 | −1.77148 | 1.00000 | −2.30505 | −1.77148 | 0.622882 | 1.00000 | 0.138156 | −2.30505 | ||||||||||||||||||
1.10 | 1.00000 | −1.74675 | 1.00000 | −3.10700 | −1.74675 | 3.74168 | 1.00000 | 0.0511396 | −3.10700 | ||||||||||||||||||
1.11 | 1.00000 | −1.70491 | 1.00000 | −0.390426 | −1.70491 | −1.43635 | 1.00000 | −0.0932983 | −0.390426 | ||||||||||||||||||
1.12 | 1.00000 | −1.50294 | 1.00000 | 1.71703 | −1.50294 | −3.21564 | 1.00000 | −0.741171 | 1.71703 | ||||||||||||||||||
1.13 | 1.00000 | −0.876341 | 1.00000 | 2.54110 | −0.876341 | −3.25278 | 1.00000 | −2.23203 | 2.54110 | ||||||||||||||||||
1.14 | 1.00000 | −0.788172 | 1.00000 | 4.07685 | −0.788172 | −1.81597 | 1.00000 | −2.37879 | 4.07685 | ||||||||||||||||||
1.15 | 1.00000 | −0.724653 | 1.00000 | −3.73538 | −0.724653 | −3.63449 | 1.00000 | −2.47488 | −3.73538 | ||||||||||||||||||
1.16 | 1.00000 | −0.466489 | 1.00000 | 1.06478 | −0.466489 | 2.04789 | 1.00000 | −2.78239 | 1.06478 | ||||||||||||||||||
1.17 | 1.00000 | −0.453821 | 1.00000 | −0.630094 | −0.453821 | 1.98767 | 1.00000 | −2.79405 | −0.630094 | ||||||||||||||||||
1.18 | 1.00000 | 0.105946 | 1.00000 | 1.78036 | 0.105946 | 0.955066 | 1.00000 | −2.98878 | 1.78036 | ||||||||||||||||||
1.19 | 1.00000 | 0.115469 | 1.00000 | −3.86988 | 0.115469 | 2.67770 | 1.00000 | −2.98667 | −3.86988 | ||||||||||||||||||
1.20 | 1.00000 | 0.232858 | 1.00000 | −1.02641 | 0.232858 | −0.709779 | 1.00000 | −2.94578 | −1.02641 | ||||||||||||||||||
See all 31 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(2003\) | \(-1\) |
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 | 4006.2.a.f | ✓ | 31 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4006.2.a.f | ✓ | 31 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{31} + 13 T_{3}^{30} + 28 T_{3}^{29} - 333 T_{3}^{28} - 1710 T_{3}^{27} + 1963 T_{3}^{26} + \cdots + 613 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4006))\).