Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1000,2,Mod(307,1000)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1000, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 2, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1000.307");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1000 = 2^{3} \cdot 5^{3} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1000.k (of order \(4\), degree \(2\), 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: | \(7.98504020213\) |
Analytic rank: | \(0\) |
Dimension: | \(64\) |
Relative dimension: | \(32\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
307.1 | −1.41419 | − | 0.00737473i | 0.633877 | − | 0.633877i | 1.99989 | + | 0.0208586i | 0 | −0.901100 | + | 0.891750i | −0.524995 | + | 0.524995i | −2.82808 | − | 0.0442468i | 2.19640i | 0 | ||||||
307.2 | −1.40990 | + | 0.110380i | 2.28932 | − | 2.28932i | 1.97563 | − | 0.311250i | 0 | −2.97502 | + | 3.48041i | 2.57361 | − | 2.57361i | −2.75109 | + | 0.656902i | − | 7.48197i | 0 | |||||
307.3 | −1.37070 | + | 0.348126i | 1.35566 | − | 1.35566i | 1.75762 | − | 0.954349i | 0 | −1.38626 | + | 2.33014i | −1.02862 | + | 1.02862i | −2.07693 | + | 1.91999i | − | 0.675627i | 0 | |||||
307.4 | −1.35196 | + | 0.414984i | −1.40270 | + | 1.40270i | 1.65558 | − | 1.12208i | 0 | 1.31429 | − | 2.47848i | −1.55425 | + | 1.55425i | −1.77262 | + | 2.20404i | − | 0.935108i | 0 | |||||
307.5 | −1.29808 | − | 0.561234i | −0.556035 | + | 0.556035i | 1.37003 | + | 1.45706i | 0 | 1.03384 | − | 0.409713i | 0.573665 | − | 0.573665i | −0.960665 | − | 2.66029i | 2.38165i | 0 | ||||||
307.6 | −1.20051 | − | 0.747512i | 1.56869 | − | 1.56869i | 0.882453 | + | 1.79479i | 0 | −3.05584 | + | 0.710615i | 2.40449 | − | 2.40449i | 0.282233 | − | 2.81431i | − | 1.92156i | 0 | |||||
307.7 | −1.19936 | + | 0.749357i | −2.28062 | + | 2.28062i | 0.876928 | − | 1.79750i | 0 | 1.02629 | − | 4.44429i | 1.63345 | − | 1.63345i | 0.295216 | + | 2.81298i | − | 7.40249i | 0 | |||||
307.8 | −1.13692 | − | 0.841074i | 1.03954 | − | 1.03954i | 0.585190 | + | 1.91247i | 0 | −2.05621 | + | 0.307547i | −2.88935 | + | 2.88935i | 0.943215 | − | 2.66652i | 0.838715i | 0 | ||||||
307.9 | −0.841074 | − | 1.13692i | 1.03954 | − | 1.03954i | −0.585190 | + | 1.91247i | 0 | −2.05621 | − | 0.307547i | 2.88935 | − | 2.88935i | 2.66652 | − | 0.943215i | 0.838715i | 0 | ||||||
307.10 | −0.749357 | + | 1.19936i | 2.28062 | − | 2.28062i | −0.876928 | − | 1.79750i | 0 | 1.02629 | + | 4.44429i | 1.63345 | − | 1.63345i | 2.81298 | + | 0.295216i | − | 7.40249i | 0 | |||||
307.11 | −0.747512 | − | 1.20051i | 1.56869 | − | 1.56869i | −0.882453 | + | 1.79479i | 0 | −3.05584 | − | 0.710615i | −2.40449 | + | 2.40449i | 2.81431 | − | 0.282233i | − | 1.92156i | 0 | |||||
307.12 | −0.561234 | − | 1.29808i | −0.556035 | + | 0.556035i | −1.37003 | + | 1.45706i | 0 | 1.03384 | + | 0.409713i | −0.573665 | + | 0.573665i | 2.66029 | + | 0.960665i | 2.38165i | 0 | ||||||
307.13 | −0.414984 | + | 1.35196i | 1.40270 | − | 1.40270i | −1.65558 | − | 1.12208i | 0 | 1.31429 | + | 2.47848i | −1.55425 | + | 1.55425i | 2.20404 | − | 1.77262i | − | 0.935108i | 0 | |||||
307.14 | −0.348126 | + | 1.37070i | −1.35566 | + | 1.35566i | −1.75762 | − | 0.954349i | 0 | −1.38626 | − | 2.33014i | −1.02862 | + | 1.02862i | 1.91999 | − | 2.07693i | − | 0.675627i | 0 | |||||
307.15 | −0.110380 | + | 1.40990i | −2.28932 | + | 2.28932i | −1.97563 | − | 0.311250i | 0 | −2.97502 | − | 3.48041i | 2.57361 | − | 2.57361i | 0.656902 | − | 2.75109i | − | 7.48197i | 0 | |||||
307.16 | −0.00737473 | − | 1.41419i | 0.633877 | − | 0.633877i | −1.99989 | + | 0.0208586i | 0 | −0.901100 | − | 0.891750i | 0.524995 | − | 0.524995i | 0.0442468 | + | 2.82808i | 2.19640i | 0 | ||||||
307.17 | 0.00737473 | + | 1.41419i | −0.633877 | + | 0.633877i | −1.99989 | + | 0.0208586i | 0 | −0.901100 | − | 0.891750i | −0.524995 | + | 0.524995i | −0.0442468 | − | 2.82808i | 2.19640i | 0 | ||||||
307.18 | 0.110380 | − | 1.40990i | 2.28932 | − | 2.28932i | −1.97563 | − | 0.311250i | 0 | −2.97502 | − | 3.48041i | −2.57361 | + | 2.57361i | −0.656902 | + | 2.75109i | − | 7.48197i | 0 | |||||
307.19 | 0.348126 | − | 1.37070i | 1.35566 | − | 1.35566i | −1.75762 | − | 0.954349i | 0 | −1.38626 | − | 2.33014i | 1.02862 | − | 1.02862i | −1.91999 | + | 2.07693i | − | 0.675627i | 0 | |||||
307.20 | 0.414984 | − | 1.35196i | −1.40270 | + | 1.40270i | −1.65558 | − | 1.12208i | 0 | 1.31429 | + | 2.47848i | 1.55425 | − | 1.55425i | −2.20404 | + | 1.77262i | − | 0.935108i | 0 | |||||
See all 64 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
5.c | odd | 4 | 2 | inner |
8.d | odd | 2 | 1 | inner |
40.e | odd | 2 | 1 | inner |
40.k | even | 4 | 2 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1000.2.k.f | ✓ | 64 |
5.b | even | 2 | 1 | inner | 1000.2.k.f | ✓ | 64 |
5.c | odd | 4 | 2 | inner | 1000.2.k.f | ✓ | 64 |
8.d | odd | 2 | 1 | inner | 1000.2.k.f | ✓ | 64 |
40.e | odd | 2 | 1 | inner | 1000.2.k.f | ✓ | 64 |
40.k | even | 4 | 2 | inner | 1000.2.k.f | ✓ | 64 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1000.2.k.f | ✓ | 64 | 1.a | even | 1 | 1 | trivial |
1000.2.k.f | ✓ | 64 | 5.b | even | 2 | 1 | inner |
1000.2.k.f | ✓ | 64 | 5.c | odd | 4 | 2 | inner |
1000.2.k.f | ✓ | 64 | 8.d | odd | 2 | 1 | inner |
1000.2.k.f | ✓ | 64 | 40.e | odd | 2 | 1 | inner |
1000.2.k.f | ✓ | 64 | 40.k | even | 4 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1000, [\chi])\):
\( T_{3}^{32} + 277 T_{3}^{28} + 25958 T_{3}^{24} + 977049 T_{3}^{20} + 16834070 T_{3}^{16} + 132564949 T_{3}^{12} + 404639833 T_{3}^{8} + 317994992 T_{3}^{4} + 69488896 \) |
\( T_{7}^{32} + 645 T_{7}^{28} + 144130 T_{7}^{24} + 13350515 T_{7}^{20} + 478100250 T_{7}^{16} + 6538914875 T_{7}^{12} + 24095051650 T_{7}^{8} + 15165478125 T_{7}^{4} + \cdots + 2562890625 \) |