Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1155,2,Mod(1121,1155)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1155, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1155.1121");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1155.l (of order \(2\), degree \(1\), minimal) |
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: | no |
Analytic conductor: | \(9.22272143346\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1121.1 | −2.79309 | 0.351991 | − | 1.69591i | 5.80135 | 1.00000i | −0.983141 | + | 4.73682i | − | 1.00000i | −10.6175 | −2.75221 | − | 1.19389i | − | 2.79309i | ||||||||||
1121.2 | −2.79309 | 0.351991 | + | 1.69591i | 5.80135 | − | 1.00000i | −0.983141 | − | 4.73682i | 1.00000i | −10.6175 | −2.75221 | + | 1.19389i | 2.79309i | |||||||||||
1121.3 | −2.55898 | −0.371837 | − | 1.69167i | 4.54836 | − | 1.00000i | 0.951521 | + | 4.32893i | 1.00000i | −6.52118 | −2.72347 | + | 1.25805i | 2.55898i | |||||||||||
1121.4 | −2.55898 | −0.371837 | + | 1.69167i | 4.54836 | 1.00000i | 0.951521 | − | 4.32893i | − | 1.00000i | −6.52118 | −2.72347 | − | 1.25805i | − | 2.55898i | ||||||||||
1121.5 | −2.21445 | 1.64596 | − | 0.539264i | 2.90381 | 1.00000i | −3.64491 | + | 1.19418i | − | 1.00000i | −2.00144 | 2.41839 | − | 1.77522i | − | 2.21445i | ||||||||||
1121.6 | −2.21445 | 1.64596 | + | 0.539264i | 2.90381 | − | 1.00000i | −3.64491 | − | 1.19418i | 1.00000i | −2.00144 | 2.41839 | + | 1.77522i | 2.21445i | |||||||||||
1121.7 | −2.04560 | −1.22443 | − | 1.22506i | 2.18447 | 1.00000i | 2.50470 | + | 2.50597i | − | 1.00000i | −0.377355 | −0.00153018 | + | 3.00000i | − | 2.04560i | ||||||||||
1121.8 | −2.04560 | −1.22443 | + | 1.22506i | 2.18447 | − | 1.00000i | 2.50470 | − | 2.50597i | 1.00000i | −0.377355 | −0.00153018 | − | 3.00000i | 2.04560i | |||||||||||
1121.9 | −1.88704 | 1.57144 | − | 0.728399i | 1.56093 | − | 1.00000i | −2.96538 | + | 1.37452i | 1.00000i | 0.828540 | 1.93887 | − | 2.28928i | 1.88704i | |||||||||||
1121.10 | −1.88704 | 1.57144 | + | 0.728399i | 1.56093 | 1.00000i | −2.96538 | − | 1.37452i | − | 1.00000i | 0.828540 | 1.93887 | + | 2.28928i | − | 1.88704i | ||||||||||
1121.11 | −1.80850 | −1.28663 | + | 1.15956i | 1.27068 | 1.00000i | 2.32688 | − | 2.09707i | − | 1.00000i | 1.31898 | 0.310844 | − | 2.98385i | − | 1.80850i | ||||||||||
1121.12 | −1.80850 | −1.28663 | − | 1.15956i | 1.27068 | − | 1.00000i | 2.32688 | + | 2.09707i | 1.00000i | 1.31898 | 0.310844 | + | 2.98385i | 1.80850i | |||||||||||
1121.13 | −1.71054 | −0.134822 | − | 1.72680i | 0.925948 | 1.00000i | 0.230619 | + | 2.95375i | − | 1.00000i | 1.83721 | −2.96365 | + | 0.465620i | − | 1.71054i | ||||||||||
1121.14 | −1.71054 | −0.134822 | + | 1.72680i | 0.925948 | − | 1.00000i | 0.230619 | − | 2.95375i | 1.00000i | 1.83721 | −2.96365 | − | 0.465620i | 1.71054i | |||||||||||
1121.15 | −0.805231 | −0.972634 | + | 1.43317i | −1.35160 | 1.00000i | 0.783195 | − | 1.15404i | − | 1.00000i | 2.69881 | −1.10797 | − | 2.78790i | − | 0.805231i | ||||||||||
1121.16 | −0.805231 | −0.972634 | − | 1.43317i | −1.35160 | − | 1.00000i | 0.783195 | + | 1.15404i | 1.00000i | 2.69881 | −1.10797 | + | 2.78790i | 0.805231i | |||||||||||
1121.17 | −0.542651 | −1.58724 | + | 0.693309i | −1.70553 | − | 1.00000i | 0.861316 | − | 0.376225i | 1.00000i | 2.01081 | 2.03865 | − | 2.20089i | 0.542651i | |||||||||||
1121.18 | −0.542651 | −1.58724 | − | 0.693309i | −1.70553 | 1.00000i | 0.861316 | + | 0.376225i | − | 1.00000i | 2.01081 | 2.03865 | + | 2.20089i | − | 0.542651i | ||||||||||
1121.19 | 0.164753 | 1.34733 | − | 1.08845i | −1.97286 | − | 1.00000i | 0.221976 | − | 0.179324i | 1.00000i | −0.654539 | 0.630571 | − | 2.93298i | − | 0.164753i | ||||||||||
1121.20 | 0.164753 | 1.34733 | + | 1.08845i | −1.97286 | 1.00000i | 0.221976 | + | 0.179324i | − | 1.00000i | −0.654539 | 0.630571 | + | 2.93298i | 0.164753i | |||||||||||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
33.d | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1155.2.l.e | ✓ | 40 |
3.b | odd | 2 | 1 | 1155.2.l.f | yes | 40 | |
11.b | odd | 2 | 1 | 1155.2.l.f | yes | 40 | |
33.d | even | 2 | 1 | inner | 1155.2.l.e | ✓ | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1155.2.l.e | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
1155.2.l.e | ✓ | 40 | 33.d | even | 2 | 1 | inner |
1155.2.l.f | yes | 40 | 3.b | odd | 2 | 1 | |
1155.2.l.f | yes | 40 | 11.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1155, [\chi])\):
\( T_{2}^{20} + 2 T_{2}^{19} - 29 T_{2}^{18} - 56 T_{2}^{17} + 347 T_{2}^{16} + 634 T_{2}^{15} - 2237 T_{2}^{14} - 3728 T_{2}^{13} + 8561 T_{2}^{12} + 12178 T_{2}^{11} - 20275 T_{2}^{10} - 21856 T_{2}^{9} + 29705 T_{2}^{8} + 19438 T_{2}^{7} + \cdots - 32 \) |
\( T_{17}^{20} - 121 T_{17}^{18} + 96 T_{17}^{17} + 5899 T_{17}^{16} - 9008 T_{17}^{15} - 146479 T_{17}^{14} + 328436 T_{17}^{13} + 1897072 T_{17}^{12} - 5804480 T_{17}^{11} - 11041120 T_{17}^{10} + 50229300 T_{17}^{9} + \cdots - 18091520 \) |