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: | \(0\) |
Dimension: | \(56\) |
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 | −2.76474 | −1.00000 | 5.64378 | −1.85589 | 2.76474 | 1.33970 | −10.0741 | 1.00000 | 5.13105 | ||||||||||||||||||
1.2 | −2.71794 | −1.00000 | 5.38720 | 2.54834 | 2.71794 | 4.22064 | −9.20622 | 1.00000 | −6.92625 | ||||||||||||||||||
1.3 | −2.65517 | −1.00000 | 5.04993 | −2.53504 | 2.65517 | 2.76662 | −8.09809 | 1.00000 | 6.73097 | ||||||||||||||||||
1.4 | −2.51234 | −1.00000 | 4.31187 | −0.437106 | 2.51234 | 0.108032 | −5.80821 | 1.00000 | 1.09816 | ||||||||||||||||||
1.5 | −2.49549 | −1.00000 | 4.22749 | 2.28895 | 2.49549 | −3.97589 | −5.55870 | 1.00000 | −5.71206 | ||||||||||||||||||
1.6 | −2.45801 | −1.00000 | 4.04182 | −3.57546 | 2.45801 | 3.37294 | −5.01881 | 1.00000 | 8.78852 | ||||||||||||||||||
1.7 | −2.36108 | −1.00000 | 3.57471 | 4.34691 | 2.36108 | −0.239132 | −3.71802 | 1.00000 | −10.2634 | ||||||||||||||||||
1.8 | −2.27923 | −1.00000 | 3.19488 | 1.61810 | 2.27923 | 2.49401 | −2.72340 | 1.00000 | −3.68801 | ||||||||||||||||||
1.9 | −2.11176 | −1.00000 | 2.45954 | −3.52613 | 2.11176 | −3.47105 | −0.970439 | 1.00000 | 7.44635 | ||||||||||||||||||
1.10 | −1.95968 | −1.00000 | 1.84033 | −3.13690 | 1.95968 | −0.0823024 | 0.312893 | 1.00000 | 6.14731 | ||||||||||||||||||
1.11 | −1.93055 | −1.00000 | 1.72702 | −2.51216 | 1.93055 | −1.67367 | 0.526999 | 1.00000 | 4.84984 | ||||||||||||||||||
1.12 | −1.84970 | −1.00000 | 1.42137 | 0.505373 | 1.84970 | −1.72444 | 1.07028 | 1.00000 | −0.934786 | ||||||||||||||||||
1.13 | −1.84393 | −1.00000 | 1.40007 | 2.87244 | 1.84393 | −1.74244 | 1.10623 | 1.00000 | −5.29656 | ||||||||||||||||||
1.14 | −1.75940 | −1.00000 | 1.09547 | −0.617389 | 1.75940 | 3.20947 | 1.59142 | 1.00000 | 1.08623 | ||||||||||||||||||
1.15 | −1.59327 | −1.00000 | 0.538495 | 2.65587 | 1.59327 | −0.355214 | 2.32857 | 1.00000 | −4.23151 | ||||||||||||||||||
1.16 | −1.50676 | −1.00000 | 0.270331 | −1.85890 | 1.50676 | 4.43102 | 2.60620 | 1.00000 | 2.80092 | ||||||||||||||||||
1.17 | −1.26654 | −1.00000 | −0.395877 | 2.14407 | 1.26654 | 1.86831 | 3.03447 | 1.00000 | −2.71555 | ||||||||||||||||||
1.18 | −1.23565 | −1.00000 | −0.473178 | −0.721310 | 1.23565 | −3.93183 | 3.05597 | 1.00000 | 0.891284 | ||||||||||||||||||
1.19 | −1.15978 | −1.00000 | −0.654916 | −0.00396327 | 1.15978 | 3.25506 | 3.07911 | 1.00000 | 0.00459651 | ||||||||||||||||||
1.20 | −0.882919 | −1.00000 | −1.22045 | −0.454468 | 0.882919 | 1.65699 | 2.84340 | 1.00000 | 0.401259 | ||||||||||||||||||
See all 56 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.g | ✓ | 56 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8007.2.a.g | ✓ | 56 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{56} - T_{2}^{55} - 86 T_{2}^{54} + 85 T_{2}^{53} + 3478 T_{2}^{52} - 3397 T_{2}^{51} + \cdots + 110519 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8007))\).