Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5915,2,Mod(1,5915)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5915, 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("5915.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 5915 = 5 \cdot 7 \cdot 13^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 5915.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: | \(47.2315127956\) |
Analytic rank: | \(1\) |
Dimension: | \(21\) |
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 | −2.74400 | −0.707717 | 5.52956 | −1.00000 | 1.94198 | −1.00000 | −9.68512 | −2.49914 | 2.74400 | ||||||||||||||||||
1.2 | −2.54977 | 1.71797 | 4.50132 | −1.00000 | −4.38043 | −1.00000 | −6.37779 | −0.0485752 | 2.54977 | ||||||||||||||||||
1.3 | −2.52854 | 3.10549 | 4.39352 | −1.00000 | −7.85236 | −1.00000 | −6.05211 | 6.64408 | 2.52854 | ||||||||||||||||||
1.4 | −2.03843 | 2.35682 | 2.15519 | −1.00000 | −4.80422 | −1.00000 | −0.316346 | 2.55462 | 2.03843 | ||||||||||||||||||
1.5 | −2.01969 | −2.51514 | 2.07915 | −1.00000 | 5.07981 | −1.00000 | −0.159865 | 3.32594 | 2.01969 | ||||||||||||||||||
1.6 | −1.55527 | −1.25293 | 0.418865 | −1.00000 | 1.94865 | −1.00000 | 2.45909 | −1.43016 | 1.55527 | ||||||||||||||||||
1.7 | −1.37365 | 1.75471 | −0.113079 | −1.00000 | −2.41036 | −1.00000 | 2.90264 | 0.0789971 | 1.37365 | ||||||||||||||||||
1.8 | −1.06682 | 3.07881 | −0.861896 | −1.00000 | −3.28453 | −1.00000 | 3.05313 | 6.47906 | 1.06682 | ||||||||||||||||||
1.9 | −0.952302 | −2.74726 | −1.09312 | −1.00000 | 2.61623 | −1.00000 | 2.94559 | 4.54746 | 0.952302 | ||||||||||||||||||
1.10 | −0.826316 | −3.03579 | −1.31720 | −1.00000 | 2.50852 | −1.00000 | 2.74106 | 6.21602 | 0.826316 | ||||||||||||||||||
1.11 | −0.449582 | −0.613470 | −1.79788 | −1.00000 | 0.275805 | −1.00000 | 1.70746 | −2.62366 | 0.449582 | ||||||||||||||||||
1.12 | 0.136571 | 0.775570 | −1.98135 | −1.00000 | 0.105921 | −1.00000 | −0.543737 | −2.39849 | −0.136571 | ||||||||||||||||||
1.13 | 0.381801 | 2.01941 | −1.85423 | −1.00000 | 0.771013 | −1.00000 | −1.47155 | 1.07803 | −0.381801 | ||||||||||||||||||
1.14 | 0.872229 | −1.78534 | −1.23922 | −1.00000 | −1.55722 | −1.00000 | −2.82534 | 0.187428 | −0.872229 | ||||||||||||||||||
1.15 | 1.05806 | −1.50254 | −0.880506 | −1.00000 | −1.58978 | −1.00000 | −3.04775 | −0.742381 | −1.05806 | ||||||||||||||||||
1.16 | 1.32241 | 2.15086 | −0.251224 | −1.00000 | 2.84433 | −1.00000 | −2.97705 | 1.62621 | −1.32241 | ||||||||||||||||||
1.17 | 1.42081 | 3.33012 | 0.0187087 | −1.00000 | 4.73148 | −1.00000 | −2.81504 | 8.08969 | −1.42081 | ||||||||||||||||||
1.18 | 1.76432 | −2.79517 | 1.11284 | −1.00000 | −4.93158 | −1.00000 | −1.56524 | 4.81295 | −1.76432 | ||||||||||||||||||
1.19 | 2.06819 | 0.524302 | 2.27742 | −1.00000 | 1.08436 | −1.00000 | 0.573763 | −2.72511 | −2.06819 | ||||||||||||||||||
1.20 | 2.53402 | 0.274221 | 4.42125 | −1.00000 | 0.694882 | −1.00000 | 6.13550 | −2.92480 | −2.53402 | ||||||||||||||||||
See all 21 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(5\) | \(1\) |
\(7\) | \(1\) |
\(13\) | \(1\) |
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 | 5915.2.a.bo | ✓ | 21 |
13.b | even | 2 | 1 | 5915.2.a.bp | yes | 21 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
5915.2.a.bo | ✓ | 21 | 1.a | even | 1 | 1 | trivial |
5915.2.a.bp | yes | 21 | 13.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5915))\):
\( T_{2}^{21} + 4 T_{2}^{20} - 23 T_{2}^{19} - 105 T_{2}^{18} + 200 T_{2}^{17} + 1137 T_{2}^{16} + \cdots + 125 \) |
\( T_{3}^{21} - 5 T_{3}^{20} - 33 T_{3}^{19} + 193 T_{3}^{18} + 397 T_{3}^{17} - 3073 T_{3}^{16} + \cdots + 8128 \) |