Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4015,2,Mod(1,4015)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4015, 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("4015.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4015 = 5 \cdot 11 \cdot 73 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4015.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: | \(32.0599364115\) |
Analytic rank: | \(1\) |
Dimension: | \(23\) |
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 | −2.71506 | 1.57103 | 5.37156 | 1.00000 | −4.26545 | 1.27925 | −9.15401 | −0.531858 | −2.71506 | ||||||||||||||||||
1.2 | −2.35657 | 2.73189 | 3.55343 | 1.00000 | −6.43790 | −2.31150 | −3.66076 | 4.46323 | −2.35657 | ||||||||||||||||||
1.3 | −2.32698 | −0.285099 | 3.41485 | 1.00000 | 0.663421 | −4.31195 | −3.29232 | −2.91872 | −2.32698 | ||||||||||||||||||
1.4 | −2.28085 | −0.867872 | 3.20226 | 1.00000 | 1.97948 | 0.729352 | −2.74217 | −2.24680 | −2.28085 | ||||||||||||||||||
1.5 | −2.06292 | 0.357225 | 2.25565 | 1.00000 | −0.736927 | 1.84250 | −0.527390 | −2.87239 | −2.06292 | ||||||||||||||||||
1.6 | −1.83447 | −2.15866 | 1.36527 | 1.00000 | 3.95999 | 0.444043 | 1.16439 | 1.65981 | −1.83447 | ||||||||||||||||||
1.7 | −1.19799 | 1.69638 | −0.564821 | 1.00000 | −2.03224 | −2.11615 | 3.07263 | −0.122309 | −1.19799 | ||||||||||||||||||
1.8 | −1.16229 | 1.36223 | −0.649085 | 1.00000 | −1.58330 | 3.38995 | 3.07900 | −1.14434 | −1.16229 | ||||||||||||||||||
1.9 | −1.09501 | −2.94583 | −0.800943 | 1.00000 | 3.22573 | 2.47152 | 3.06707 | 5.67794 | −1.09501 | ||||||||||||||||||
1.10 | −1.08357 | −1.29689 | −0.825886 | 1.00000 | 1.40527 | −5.06508 | 3.06203 | −1.31806 | −1.08357 | ||||||||||||||||||
1.11 | −0.654679 | 2.96699 | −1.57140 | 1.00000 | −1.94243 | −1.18438 | 2.33812 | 5.80304 | −0.654679 | ||||||||||||||||||
1.12 | −0.182665 | 1.60195 | −1.96663 | 1.00000 | −0.292620 | −3.46851 | 0.724566 | −0.433762 | −0.182665 | ||||||||||||||||||
1.13 | 0.152616 | −0.242697 | −1.97671 | 1.00000 | −0.0370393 | 2.57515 | −0.606909 | −2.94110 | 0.152616 | ||||||||||||||||||
1.14 | 0.152825 | −1.31847 | −1.97664 | 1.00000 | −0.201495 | −0.959999 | −0.607731 | −1.26164 | 0.152825 | ||||||||||||||||||
1.15 | 0.430868 | −1.25921 | −1.81435 | 1.00000 | −0.542554 | 4.34543 | −1.64348 | −1.41438 | 0.430868 | ||||||||||||||||||
1.16 | 0.772478 | −3.03398 | −1.40328 | 1.00000 | −2.34368 | 1.44909 | −2.62896 | 6.20504 | 0.772478 | ||||||||||||||||||
1.17 | 1.18208 | −1.23608 | −0.602686 | 1.00000 | −1.46115 | −1.06876 | −3.07658 | −1.47210 | 1.18208 | ||||||||||||||||||
1.18 | 1.32991 | 2.72450 | −0.231346 | 1.00000 | 3.62333 | −4.53918 | −2.96748 | 4.42288 | 1.32991 | ||||||||||||||||||
1.19 | 1.55677 | 0.976240 | 0.423526 | 1.00000 | 1.51978 | −0.0535957 | −2.45420 | −2.04696 | 1.55677 | ||||||||||||||||||
1.20 | 1.79465 | 0.618778 | 1.22075 | 1.00000 | 1.11049 | 1.73339 | −1.39847 | −2.61711 | 1.79465 | ||||||||||||||||||
See all 23 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(5\) | \(-1\) |
\(11\) | \(-1\) |
\(73\) | \(1\) |
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 | 4015.2.a.b | ✓ | 23 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4015.2.a.b | ✓ | 23 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{23} + 5 T_{2}^{22} - 18 T_{2}^{21} - 121 T_{2}^{20} + 95 T_{2}^{19} + 1226 T_{2}^{18} + 170 T_{2}^{17} + \cdots - 9 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4015))\).