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: | \(0\) |
Dimension: | \(27\) |
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 | −2.34661 | 2.98114 | 3.50659 | −1.00000 | −6.99558 | 0.286679 | −3.53537 | 5.88720 | 2.34661 | ||||||||||||||||||
1.2 | −2.33465 | −0.983408 | 3.45061 | −1.00000 | 2.29592 | −2.48043 | −3.38668 | −2.03291 | 2.33465 | ||||||||||||||||||
1.3 | −2.24746 | 1.22256 | 3.05108 | −1.00000 | −2.74765 | −0.859995 | −2.36225 | −1.50535 | 2.24746 | ||||||||||||||||||
1.4 | −2.19462 | −2.47425 | 2.81638 | −1.00000 | 5.43004 | 3.64013 | −1.79164 | 3.12189 | 2.19462 | ||||||||||||||||||
1.5 | −1.64001 | 1.19849 | 0.689644 | −1.00000 | −1.96554 | −2.90687 | 2.14900 | −1.56361 | 1.64001 | ||||||||||||||||||
1.6 | −1.49741 | 2.11504 | 0.242240 | −1.00000 | −3.16709 | 4.23699 | 2.63209 | 1.47340 | 1.49741 | ||||||||||||||||||
1.7 | −1.35811 | 0.129284 | −0.155539 | −1.00000 | −0.175582 | −0.681246 | 2.92746 | −2.98329 | 1.35811 | ||||||||||||||||||
1.8 | −1.32466 | −2.16018 | −0.245282 | −1.00000 | 2.86150 | −3.08475 | 2.97423 | 1.66638 | 1.32466 | ||||||||||||||||||
1.9 | −0.788550 | −0.398652 | −1.37819 | −1.00000 | 0.314357 | 4.29459 | 2.66387 | −2.84108 | 0.788550 | ||||||||||||||||||
1.10 | −0.590698 | 0.738483 | −1.65108 | −1.00000 | −0.436221 | −1.02669 | 2.15668 | −2.45464 | 0.590698 | ||||||||||||||||||
1.11 | −0.339834 | 2.87141 | −1.88451 | −1.00000 | −0.975803 | 0.914353 | 1.32009 | 5.24498 | 0.339834 | ||||||||||||||||||
1.12 | −0.0879542 | −2.14989 | −1.99226 | −1.00000 | 0.189092 | 3.99167 | 0.351137 | 1.62202 | 0.0879542 | ||||||||||||||||||
1.13 | −0.0560878 | −2.46183 | −1.99685 | −1.00000 | 0.138079 | −2.79584 | 0.224175 | 3.06062 | 0.0560878 | ||||||||||||||||||
1.14 | 0.0645883 | −2.75103 | −1.99583 | −1.00000 | −0.177684 | 0.602441 | −0.258084 | 4.56815 | −0.0645883 | ||||||||||||||||||
1.15 | 0.782969 | −1.17473 | −1.38696 | −1.00000 | −0.919776 | 1.16812 | −2.65188 | −1.62001 | −0.782969 | ||||||||||||||||||
1.16 | 0.924358 | 1.45154 | −1.14556 | −1.00000 | 1.34174 | 3.41942 | −2.90763 | −0.893034 | −0.924358 | ||||||||||||||||||
1.17 | 0.966384 | 2.94159 | −1.06610 | −1.00000 | 2.84271 | −4.68182 | −2.96303 | 5.65296 | −0.966384 | ||||||||||||||||||
1.18 | 1.13591 | 3.01832 | −0.709697 | −1.00000 | 3.42856 | 3.60395 | −3.07799 | 6.11027 | −1.13591 | ||||||||||||||||||
1.19 | 1.36344 | −0.755926 | −0.141019 | −1.00000 | −1.03066 | 1.53902 | −2.91916 | −2.42858 | −1.36344 | ||||||||||||||||||
1.20 | 1.39893 | −0.528085 | −0.0429944 | −1.00000 | −0.738755 | −3.55790 | −2.85801 | −2.72113 | −1.39893 | ||||||||||||||||||
See all 27 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.e | ✓ | 27 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4015.2.a.e | ✓ | 27 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{27} - 6 T_{2}^{26} - 20 T_{2}^{25} + 179 T_{2}^{24} + 80 T_{2}^{23} - 2285 T_{2}^{22} + 1258 T_{2}^{21} + \cdots + 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4015))\).