Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [750,2,Mod(107,750)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(750, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([10, 13]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("750.107");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 750 = 2 \cdot 3 \cdot 5^{3} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 750.l (of order \(20\), degree \(8\), not 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: | \(5.98878015160\) |
Analytic rank: | \(0\) |
Dimension: | \(80\) |
Relative dimension: | \(10\) over \(\Q(\zeta_{20})\) |
Twist minimal: | no (minimal twist has level 150) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{20}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
107.1 | −0.891007 | − | 0.453990i | −1.73204 | − | 0.00452789i | 0.587785 | + | 0.809017i | 0 | 1.54121 | + | 0.790366i | 0.152718 | + | 0.152718i | −0.156434 | − | 0.987688i | 2.99996 | + | 0.0156850i | 0 | ||||
107.2 | −0.891007 | − | 0.453990i | −0.832356 | + | 1.51894i | 0.587785 | + | 0.809017i | 0 | 1.43122 | − | 0.975505i | −0.0556476 | − | 0.0556476i | −0.156434 | − | 0.987688i | −1.61437 | − | 2.52860i | 0 | ||||
107.3 | −0.891007 | − | 0.453990i | −0.368756 | − | 1.69234i | 0.587785 | + | 0.809017i | 0 | −0.439743 | + | 1.67530i | 2.72680 | + | 2.72680i | −0.156434 | − | 0.987688i | −2.72804 | + | 1.24812i | 0 | ||||
107.4 | −0.891007 | − | 0.453990i | 1.37558 | − | 1.05251i | 0.587785 | + | 0.809017i | 0 | −1.70348 | + | 0.313291i | −0.462249 | − | 0.462249i | −0.156434 | − | 0.987688i | 0.784450 | − | 2.89562i | 0 | ||||
107.5 | −0.891007 | − | 0.453990i | 1.69961 | + | 0.333634i | 0.587785 | + | 0.809017i | 0 | −1.36290 | − | 1.06888i | −2.58285 | − | 2.58285i | −0.156434 | − | 0.987688i | 2.77738 | + | 1.13410i | 0 | ||||
107.6 | 0.891007 | + | 0.453990i | −1.71953 | − | 0.207905i | 0.587785 | + | 0.809017i | 0 | −1.43772 | − | 0.965894i | −2.58285 | − | 2.58285i | 0.156434 | + | 0.987688i | 2.91355 | + | 0.714995i | 0 | ||||
107.7 | 0.891007 | + | 0.453990i | −0.983013 | − | 1.42607i | 0.587785 | + | 0.809017i | 0 | −0.228447 | − | 1.71692i | −0.462249 | − | 0.462249i | 0.156434 | + | 0.987688i | −1.06737 | + | 2.80370i | 0 | ||||
107.8 | 0.891007 | + | 0.453990i | 0.322239 | + | 1.70181i | 0.587785 | + | 0.809017i | 0 | −0.485490 | + | 1.66262i | −0.0556476 | − | 0.0556476i | 0.156434 | + | 0.987688i | −2.79232 | + | 1.09678i | 0 | ||||
107.9 | 0.891007 | + | 0.453990i | 0.873670 | − | 1.49556i | 0.587785 | + | 0.809017i | 0 | 1.45742 | − | 0.935916i | 2.72680 | + | 2.72680i | 0.156434 | + | 0.987688i | −1.47340 | − | 2.61325i | 0 | ||||
107.10 | 0.891007 | + | 0.453990i | 1.64867 | + | 0.530925i | 0.587785 | + | 0.809017i | 0 | 1.22794 | + | 1.22154i | 0.152718 | + | 0.152718i | 0.156434 | + | 0.987688i | 2.43624 | + | 1.75064i | 0 | ||||
143.1 | −0.453990 | + | 0.891007i | −1.72904 | + | 0.102145i | −0.587785 | − | 0.809017i | 0 | 0.693954 | − | 1.58696i | 2.97677 | − | 2.97677i | 0.987688 | − | 0.156434i | 2.97913 | − | 0.353226i | 0 | ||||
143.2 | −0.453990 | + | 0.891007i | −1.60692 | + | 0.646378i | −0.587785 | − | 0.809017i | 0 | 0.153600 | − | 1.72523i | −2.03922 | + | 2.03922i | 0.987688 | − | 0.156434i | 2.16439 | − | 2.07736i | 0 | ||||
143.3 | −0.453990 | + | 0.891007i | −0.522656 | − | 1.65131i | −0.587785 | − | 0.809017i | 0 | 1.70861 | + | 0.283990i | −0.712495 | + | 0.712495i | 0.987688 | − | 0.156434i | −2.45366 | + | 1.72614i | 0 | ||||
143.4 | −0.453990 | + | 0.891007i | 0.666690 | + | 1.59860i | −0.587785 | − | 0.809017i | 0 | −1.72703 | − | 0.131724i | 1.51403 | − | 1.51403i | 0.987688 | − | 0.156434i | −2.11105 | + | 2.13154i | 0 | ||||
143.5 | −0.453990 | + | 0.891007i | 1.43185 | − | 0.974581i | −0.587785 | − | 0.809017i | 0 | 0.218312 | + | 1.71824i | −3.13589 | + | 3.13589i | 0.987688 | − | 0.156434i | 1.10038 | − | 2.79091i | 0 | ||||
143.6 | 0.453990 | − | 0.891007i | −1.61285 | − | 0.631448i | −0.587785 | − | 0.809017i | 0 | −1.29484 | + | 1.15039i | 2.97677 | − | 2.97677i | −0.987688 | + | 0.156434i | 2.20255 | + | 2.03686i | 0 | ||||
143.7 | 0.453990 | − | 0.891007i | −1.32853 | − | 1.11131i | −0.587785 | − | 0.809017i | 0 | −1.59332 | + | 0.679206i | −2.03922 | + | 2.03922i | −0.987688 | + | 0.156434i | 0.529988 | + | 2.95281i | 0 | ||||
143.8 | 0.453990 | − | 0.891007i | −1.00736 | + | 1.40898i | −0.587785 | − | 0.809017i | 0 | 0.798080 | + | 1.53723i | −0.712495 | + | 0.712495i | −0.987688 | + | 0.156434i | −0.970457 | − | 2.83870i | 0 | ||||
143.9 | 0.453990 | − | 0.891007i | 1.06061 | + | 1.36935i | −0.587785 | − | 0.809017i | 0 | 1.70160 | − | 0.323338i | −3.13589 | + | 3.13589i | −0.987688 | + | 0.156434i | −0.750223 | + | 2.90468i | 0 | ||||
143.10 | 0.453990 | − | 0.891007i | 1.12805 | − | 1.31434i | −0.587785 | − | 0.809017i | 0 | −0.658960 | − | 1.60180i | 1.51403 | − | 1.51403i | −0.987688 | + | 0.156434i | −0.454984 | − | 2.96530i | 0 | ||||
See all 80 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
25.f | odd | 20 | 1 | inner |
75.l | even | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 750.2.l.b | 80 | |
3.b | odd | 2 | 1 | inner | 750.2.l.b | 80 | |
5.b | even | 2 | 1 | 150.2.l.a | ✓ | 80 | |
5.c | odd | 4 | 1 | 750.2.l.a | 80 | ||
5.c | odd | 4 | 1 | 750.2.l.c | 80 | ||
15.d | odd | 2 | 1 | 150.2.l.a | ✓ | 80 | |
15.e | even | 4 | 1 | 750.2.l.a | 80 | ||
15.e | even | 4 | 1 | 750.2.l.c | 80 | ||
25.d | even | 5 | 1 | 750.2.l.a | 80 | ||
25.e | even | 10 | 1 | 750.2.l.c | 80 | ||
25.f | odd | 20 | 1 | 150.2.l.a | ✓ | 80 | |
25.f | odd | 20 | 1 | inner | 750.2.l.b | 80 | |
75.h | odd | 10 | 1 | 750.2.l.c | 80 | ||
75.j | odd | 10 | 1 | 750.2.l.a | 80 | ||
75.l | even | 20 | 1 | 150.2.l.a | ✓ | 80 | |
75.l | even | 20 | 1 | inner | 750.2.l.b | 80 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
150.2.l.a | ✓ | 80 | 5.b | even | 2 | 1 | |
150.2.l.a | ✓ | 80 | 15.d | odd | 2 | 1 | |
150.2.l.a | ✓ | 80 | 25.f | odd | 20 | 1 | |
150.2.l.a | ✓ | 80 | 75.l | even | 20 | 1 | |
750.2.l.a | 80 | 5.c | odd | 4 | 1 | ||
750.2.l.a | 80 | 15.e | even | 4 | 1 | ||
750.2.l.a | 80 | 25.d | even | 5 | 1 | ||
750.2.l.a | 80 | 75.j | odd | 10 | 1 | ||
750.2.l.b | 80 | 1.a | even | 1 | 1 | trivial | |
750.2.l.b | 80 | 3.b | odd | 2 | 1 | inner | |
750.2.l.b | 80 | 25.f | odd | 20 | 1 | inner | |
750.2.l.b | 80 | 75.l | even | 20 | 1 | inner | |
750.2.l.c | 80 | 5.c | odd | 4 | 1 | ||
750.2.l.c | 80 | 15.e | even | 4 | 1 | ||
750.2.l.c | 80 | 25.e | even | 10 | 1 | ||
750.2.l.c | 80 | 75.h | odd | 10 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(750, [\chi])\):
\( T_{7}^{40} + 2 T_{7}^{39} + 2 T_{7}^{38} - 20 T_{7}^{37} + 1183 T_{7}^{36} + 2404 T_{7}^{35} + \cdots + 5776 \) |
\( T_{13}^{40} + 20 T_{13}^{38} + 180 T_{13}^{37} - 735 T_{13}^{36} - 600 T_{13}^{35} + \cdots + 14\!\cdots\!00 \) |