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: | \(32\) |
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 | −2.80101 | 2.07520 | 5.84563 | −1.00000 | −5.81265 | 2.07490 | −10.7716 | 1.30645 | 2.80101 | ||||||||||||||||||
1.2 | −2.75666 | −2.97515 | 5.59919 | −1.00000 | 8.20149 | 1.32848 | −9.92176 | 5.85152 | 2.75666 | ||||||||||||||||||
1.3 | −2.62713 | −1.17329 | 4.90182 | −1.00000 | 3.08238 | 3.02645 | −7.62347 | −1.62339 | 2.62713 | ||||||||||||||||||
1.4 | −2.55668 | 0.570205 | 4.53663 | −1.00000 | −1.45783 | −4.65776 | −6.48536 | −2.67487 | 2.55668 | ||||||||||||||||||
1.5 | −2.31178 | −1.68436 | 3.34432 | −1.00000 | 3.89386 | −2.49479 | −3.10778 | −0.162937 | 2.31178 | ||||||||||||||||||
1.6 | −2.06358 | 2.59950 | 2.25838 | −1.00000 | −5.36430 | −1.01430 | −0.533185 | 3.75743 | 2.06358 | ||||||||||||||||||
1.7 | −2.01115 | −3.15685 | 2.04473 | −1.00000 | 6.34891 | −3.03904 | −0.0899564 | 6.96573 | 2.01115 | ||||||||||||||||||
1.8 | −1.96967 | 2.18511 | 1.87958 | −1.00000 | −4.30394 | 1.92639 | 0.237186 | 1.77472 | 1.96967 | ||||||||||||||||||
1.9 | −1.75190 | −0.429918 | 1.06914 | −1.00000 | 0.753172 | 3.79529 | 1.63077 | −2.81517 | 1.75190 | ||||||||||||||||||
1.10 | −1.71308 | 1.21665 | 0.934626 | −1.00000 | −2.08421 | −4.29501 | 1.82507 | −1.51977 | 1.71308 | ||||||||||||||||||
1.11 | −1.50747 | −0.655267 | 0.272451 | −1.00000 | 0.987792 | −1.30717 | 2.60422 | −2.57063 | 1.50747 | ||||||||||||||||||
1.12 | −0.962434 | 2.32544 | −1.07372 | −1.00000 | −2.23808 | 2.98112 | 2.95825 | 2.40766 | 0.962434 | ||||||||||||||||||
1.13 | −0.960547 | −2.83486 | −1.07735 | −1.00000 | 2.72301 | 4.60653 | 2.95594 | 5.03641 | 0.960547 | ||||||||||||||||||
1.14 | −0.763072 | 0.889359 | −1.41772 | −1.00000 | −0.678645 | 2.40954 | 2.60797 | −2.20904 | 0.763072 | ||||||||||||||||||
1.15 | −0.711670 | −2.33630 | −1.49353 | −1.00000 | 1.66268 | −3.28922 | 2.48624 | 2.45831 | 0.711670 | ||||||||||||||||||
1.16 | −0.193228 | −1.07555 | −1.96266 | −1.00000 | 0.207825 | 1.51348 | 0.765696 | −1.84320 | 0.193228 | ||||||||||||||||||
1.17 | 0.0852596 | 0.824276 | −1.99273 | −1.00000 | 0.0702774 | 0.193385 | −0.340419 | −2.32057 | −0.0852596 | ||||||||||||||||||
1.18 | 0.338336 | −0.770870 | −1.88553 | −1.00000 | −0.260813 | −3.60696 | −1.31461 | −2.40576 | −0.338336 | ||||||||||||||||||
1.19 | 0.467946 | 3.03596 | −1.78103 | −1.00000 | 1.42067 | −2.71430 | −1.76932 | 6.21706 | −0.467946 | ||||||||||||||||||
1.20 | 0.472050 | −2.02631 | −1.77717 | −1.00000 | −0.956522 | −4.07098 | −1.78301 | 1.10595 | −0.472050 | ||||||||||||||||||
See all 32 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.g | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4015.2.a.g | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 5 T_{2}^{31} - 38 T_{2}^{30} - 219 T_{2}^{29} + 602 T_{2}^{28} + 4267 T_{2}^{27} + \cdots + 1024 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4015))\).