Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6015,2,Mod(1,6015)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6015, 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("6015.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6015 = 3 \cdot 5 \cdot 401 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6015.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: | \(48.0300168158\) |
Analytic rank: | \(1\) |
Dimension: | \(28\) |
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 | −2.60633 | 1.00000 | 4.79297 | −1.00000 | −2.60633 | −3.84306 | −7.27940 | 1.00000 | 2.60633 | ||||||||||||||||||
1.2 | −2.51673 | 1.00000 | 4.33393 | −1.00000 | −2.51673 | −0.0213252 | −5.87387 | 1.00000 | 2.51673 | ||||||||||||||||||
1.3 | −2.31182 | 1.00000 | 3.34451 | −1.00000 | −2.31182 | −2.77511 | −3.10826 | 1.00000 | 2.31182 | ||||||||||||||||||
1.4 | −2.11452 | 1.00000 | 2.47118 | −1.00000 | −2.11452 | 1.95178 | −0.996316 | 1.00000 | 2.11452 | ||||||||||||||||||
1.5 | −1.92652 | 1.00000 | 1.71147 | −1.00000 | −1.92652 | 1.15268 | 0.555850 | 1.00000 | 1.92652 | ||||||||||||||||||
1.6 | −1.85217 | 1.00000 | 1.43054 | −1.00000 | −1.85217 | −4.50663 | 1.05474 | 1.00000 | 1.85217 | ||||||||||||||||||
1.7 | −1.62087 | 1.00000 | 0.627218 | −1.00000 | −1.62087 | 2.78454 | 2.22510 | 1.00000 | 1.62087 | ||||||||||||||||||
1.8 | −1.59553 | 1.00000 | 0.545711 | −1.00000 | −1.59553 | −0.379773 | 2.32036 | 1.00000 | 1.59553 | ||||||||||||||||||
1.9 | −1.45436 | 1.00000 | 0.115161 | −1.00000 | −1.45436 | −2.79967 | 2.74123 | 1.00000 | 1.45436 | ||||||||||||||||||
1.10 | −0.932405 | 1.00000 | −1.13062 | −1.00000 | −0.932405 | 3.48092 | 2.91901 | 1.00000 | 0.932405 | ||||||||||||||||||
1.11 | −0.711184 | 1.00000 | −1.49422 | −1.00000 | −0.711184 | −4.85706 | 2.48503 | 1.00000 | 0.711184 | ||||||||||||||||||
1.12 | −0.549158 | 1.00000 | −1.69842 | −1.00000 | −0.549158 | 0.129960 | 2.03102 | 1.00000 | 0.549158 | ||||||||||||||||||
1.13 | −0.517037 | 1.00000 | −1.73267 | −1.00000 | −0.517037 | 1.87988 | 1.92993 | 1.00000 | 0.517037 | ||||||||||||||||||
1.14 | −0.327847 | 1.00000 | −1.89252 | −1.00000 | −0.327847 | −1.80659 | 1.27615 | 1.00000 | 0.327847 | ||||||||||||||||||
1.15 | 0.113161 | 1.00000 | −1.98719 | −1.00000 | 0.113161 | 2.66791 | −0.451194 | 1.00000 | −0.113161 | ||||||||||||||||||
1.16 | 0.344505 | 1.00000 | −1.88132 | −1.00000 | 0.344505 | −3.88123 | −1.33713 | 1.00000 | −0.344505 | ||||||||||||||||||
1.17 | 0.403903 | 1.00000 | −1.83686 | −1.00000 | 0.403903 | 0.330671 | −1.54972 | 1.00000 | −0.403903 | ||||||||||||||||||
1.18 | 0.886913 | 1.00000 | −1.21339 | −1.00000 | 0.886913 | 2.28245 | −2.84999 | 1.00000 | −0.886913 | ||||||||||||||||||
1.19 | 0.894188 | 1.00000 | −1.20043 | −1.00000 | 0.894188 | −1.23837 | −2.86178 | 1.00000 | −0.894188 | ||||||||||||||||||
1.20 | 0.902444 | 1.00000 | −1.18560 | −1.00000 | 0.902444 | −2.02428 | −2.87482 | 1.00000 | −0.902444 | ||||||||||||||||||
See all 28 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
\(5\) | \(1\) |
\(401\) | \(-1\) |
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 | 6015.2.a.c | ✓ | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6015.2.a.c | ✓ | 28 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} + T_{2}^{27} - 38 T_{2}^{26} - 37 T_{2}^{25} + 636 T_{2}^{24} + 602 T_{2}^{23} - 6175 T_{2}^{22} + \cdots + 88 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6015))\).