Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8015,2,Mod(1,8015)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8015, 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("8015.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8015 = 5 \cdot 7 \cdot 229 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8015.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: | \(64.0000972201\) |
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 | −2.79746 | −0.603519 | 5.82580 | 1.00000 | 1.68832 | −1.00000 | −10.7025 | −2.63576 | −2.79746 | ||||||||||||||||||
1.2 | −2.53948 | 2.62786 | 4.44894 | 1.00000 | −6.67339 | −1.00000 | −6.21901 | 3.90566 | −2.53948 | ||||||||||||||||||
1.3 | −2.32161 | −1.61048 | 3.38989 | 1.00000 | 3.73891 | −1.00000 | −3.22677 | −0.406353 | −2.32161 | ||||||||||||||||||
1.4 | −2.27226 | −1.26810 | 3.16315 | 1.00000 | 2.88144 | −1.00000 | −2.64298 | −1.39193 | −2.27226 | ||||||||||||||||||
1.5 | −2.26027 | 2.68549 | 3.10882 | 1.00000 | −6.06994 | −1.00000 | −2.50623 | 4.21188 | −2.26027 | ||||||||||||||||||
1.6 | −2.24100 | −0.455413 | 3.02210 | 1.00000 | 1.02058 | −1.00000 | −2.29052 | −2.79260 | −2.24100 | ||||||||||||||||||
1.7 | −2.23576 | 0.480979 | 2.99861 | 1.00000 | −1.07535 | −1.00000 | −2.23266 | −2.76866 | −2.23576 | ||||||||||||||||||
1.8 | −1.87978 | 3.08968 | 1.53356 | 1.00000 | −5.80791 | −1.00000 | 0.876807 | 6.54615 | −1.87978 | ||||||||||||||||||
1.9 | −1.84251 | −2.60277 | 1.39485 | 1.00000 | 4.79563 | −1.00000 | 1.11499 | 3.77440 | −1.84251 | ||||||||||||||||||
1.10 | −1.65863 | 1.21505 | 0.751045 | 1.00000 | −2.01532 | −1.00000 | 2.07155 | −1.52365 | −1.65863 | ||||||||||||||||||
1.11 | −1.54745 | −2.71325 | 0.394610 | 1.00000 | 4.19863 | −1.00000 | 2.48426 | 4.36174 | −1.54745 | ||||||||||||||||||
1.12 | −1.41331 | 0.0185355 | −0.00254977 | 1.00000 | −0.0261965 | −1.00000 | 2.83023 | −2.99966 | −1.41331 | ||||||||||||||||||
1.13 | −1.25477 | 1.22958 | −0.425541 | 1.00000 | −1.54284 | −1.00000 | 3.04351 | −1.48814 | −1.25477 | ||||||||||||||||||
1.14 | −1.22585 | 2.00155 | −0.497288 | 1.00000 | −2.45360 | −1.00000 | 3.06130 | 1.00619 | −1.22585 | ||||||||||||||||||
1.15 | −1.22552 | −2.22296 | −0.498095 | 1.00000 | 2.72428 | −1.00000 | 3.06147 | 1.94153 | −1.22552 | ||||||||||||||||||
1.16 | −1.03648 | −2.16444 | −0.925705 | 1.00000 | 2.24341 | −1.00000 | 3.03244 | 1.68482 | −1.03648 | ||||||||||||||||||
1.17 | −0.631186 | 1.05743 | −1.60160 | 1.00000 | −0.667437 | −1.00000 | 2.27328 | −1.88184 | −0.631186 | ||||||||||||||||||
1.18 | −0.612476 | −1.57448 | −1.62487 | 1.00000 | 0.964330 | −1.00000 | 2.22015 | −0.521024 | −0.612476 | ||||||||||||||||||
1.19 | −0.583301 | −0.502435 | −1.65976 | 1.00000 | 0.293071 | −1.00000 | 2.13474 | −2.74756 | −0.583301 | ||||||||||||||||||
1.20 | −0.508100 | 2.08594 | −1.74183 | 1.00000 | −1.05987 | −1.00000 | 1.90123 | 1.35113 | −0.508100 | ||||||||||||||||||
See all 44 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(5\) | \(-1\) |
\(7\) | \(1\) |
\(229\) | \(-1\) |
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 | 8015.2.a.i | ✓ | 44 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8015.2.a.i | ✓ | 44 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8015))\):
\( T_{2}^{44} + 2 T_{2}^{43} - 57 T_{2}^{42} - 113 T_{2}^{41} + 1495 T_{2}^{40} + 2937 T_{2}^{39} + \cdots - 583 \) |
\( T_{3}^{44} - 74 T_{3}^{42} - 2 T_{3}^{41} + 2522 T_{3}^{40} + 131 T_{3}^{39} - 52563 T_{3}^{38} + \cdots + 2560 \) |