Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8007,2,Mod(1,8007)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8007, 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("8007.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8007 = 3 \cdot 17 \cdot 157 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8007.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: | \(63.9362168984\) |
Analytic rank: | \(1\) |
Dimension: | \(48\) |
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 | −2.78117 | −1.00000 | 5.73492 | 2.04412 | 2.78117 | −0.793043 | −10.3875 | 1.00000 | −5.68506 | ||||||||||||||||||
1.2 | −2.66751 | −1.00000 | 5.11563 | −0.759881 | 2.66751 | −3.14723 | −8.31098 | 1.00000 | 2.02699 | ||||||||||||||||||
1.3 | −2.52044 | −1.00000 | 4.35263 | −1.87454 | 2.52044 | −3.72156 | −5.92968 | 1.00000 | 4.72466 | ||||||||||||||||||
1.4 | −2.46582 | −1.00000 | 4.08028 | 3.93570 | 2.46582 | −1.94268 | −5.12960 | 1.00000 | −9.70474 | ||||||||||||||||||
1.5 | −2.39228 | −1.00000 | 3.72299 | 0.325110 | 2.39228 | −0.363079 | −4.12187 | 1.00000 | −0.777753 | ||||||||||||||||||
1.6 | −2.28381 | −1.00000 | 3.21580 | 1.64317 | 2.28381 | 3.93496 | −2.77667 | 1.00000 | −3.75270 | ||||||||||||||||||
1.7 | −2.27518 | −1.00000 | 3.17643 | 2.61308 | 2.27518 | 3.54577 | −2.67658 | 1.00000 | −5.94522 | ||||||||||||||||||
1.8 | −2.17906 | −1.00000 | 2.74829 | −2.11197 | 2.17906 | 2.69680 | −1.63058 | 1.00000 | 4.60211 | ||||||||||||||||||
1.9 | −2.00652 | −1.00000 | 2.02613 | −1.83243 | 2.00652 | −1.92586 | −0.0524220 | 1.00000 | 3.67682 | ||||||||||||||||||
1.10 | −1.95217 | −1.00000 | 1.81095 | −1.54543 | 1.95217 | 2.71405 | 0.369057 | 1.00000 | 3.01694 | ||||||||||||||||||
1.11 | −1.70660 | −1.00000 | 0.912495 | 1.58325 | 1.70660 | −2.98844 | 1.85594 | 1.00000 | −2.70198 | ||||||||||||||||||
1.12 | −1.54663 | −1.00000 | 0.392054 | 3.59105 | 1.54663 | −3.93711 | 2.48689 | 1.00000 | −5.55401 | ||||||||||||||||||
1.13 | −1.50666 | −1.00000 | 0.270014 | −0.475693 | 1.50666 | −0.891979 | 2.60649 | 1.00000 | 0.716706 | ||||||||||||||||||
1.14 | −1.44064 | −1.00000 | 0.0754557 | −3.22452 | 1.44064 | 1.08354 | 2.77258 | 1.00000 | 4.64538 | ||||||||||||||||||
1.15 | −1.38255 | −1.00000 | −0.0885651 | 2.12725 | 1.38255 | −0.203962 | 2.88754 | 1.00000 | −2.94103 | ||||||||||||||||||
1.16 | −1.24727 | −1.00000 | −0.444326 | −3.92446 | 1.24727 | −1.90187 | 3.04873 | 1.00000 | 4.89484 | ||||||||||||||||||
1.17 | −0.980818 | −1.00000 | −1.03800 | 1.00436 | 0.980818 | −3.53145 | 2.97972 | 1.00000 | −0.985094 | ||||||||||||||||||
1.18 | −0.901374 | −1.00000 | −1.18752 | 1.97657 | 0.901374 | 4.00403 | 2.87315 | 1.00000 | −1.78163 | ||||||||||||||||||
1.19 | −0.857233 | −1.00000 | −1.26515 | 0.241794 | 0.857233 | −3.40893 | 2.79900 | 1.00000 | −0.207274 | ||||||||||||||||||
1.20 | −0.557584 | −1.00000 | −1.68910 | 2.57031 | 0.557584 | 2.28117 | 2.05698 | 1.00000 | −1.43316 | ||||||||||||||||||
See all 48 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(17\) | \(1\) |
\(157\) | \(1\) |
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 | 8007.2.a.f | ✓ | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8007.2.a.f | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} + T_{2}^{47} - 70 T_{2}^{46} - 67 T_{2}^{45} + 2280 T_{2}^{44} + 2081 T_{2}^{43} - 45907 T_{2}^{42} + \cdots + 787 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8007))\).