Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [125,3,Mod(7,125)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(125, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([17]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("125.7");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 125 = 5^{3} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 125.f (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: | \(3.40600330450\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{20})\) |
Twist minimal: | no (minimal twist has level 25) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
7.1 | −2.70026 | + | 1.37585i | 4.42692 | − | 0.701156i | 3.04732 | − | 4.19427i | 0 | −10.9892 | + | 7.98410i | 4.77540 | + | 4.77540i | −0.561510 | + | 3.54523i | 10.5465 | − | 3.42677i | 0 | ||||
7.2 | −0.972743 | + | 0.495637i | −0.872241 | + | 0.138149i | −1.65057 | + | 2.27181i | 0 | 0.779995 | − | 0.566699i | −1.62783 | − | 1.62783i | 1.16272 | − | 7.34115i | −7.81779 | + | 2.54015i | 0 | ||||
7.3 | 1.61837 | − | 0.824603i | −3.42034 | + | 0.541729i | −0.411975 | + | 0.567034i | 0 | −5.08868 | + | 3.69715i | 8.06323 | + | 8.06323i | −1.33571 | + | 8.43332i | 2.84577 | − | 0.924645i | 0 | ||||
7.4 | 2.38234 | − | 1.21387i | 3.57679 | − | 0.566508i | 1.85096 | − | 2.54762i | 0 | 7.83348 | − | 5.69136i | −6.54971 | − | 6.54971i | −0.355933 | + | 2.24727i | 3.91299 | − | 1.27141i | 0 | ||||
18.1 | −2.70026 | − | 1.37585i | 4.42692 | + | 0.701156i | 3.04732 | + | 4.19427i | 0 | −10.9892 | − | 7.98410i | 4.77540 | − | 4.77540i | −0.561510 | − | 3.54523i | 10.5465 | + | 3.42677i | 0 | ||||
18.2 | −0.972743 | − | 0.495637i | −0.872241 | − | 0.138149i | −1.65057 | − | 2.27181i | 0 | 0.779995 | + | 0.566699i | −1.62783 | + | 1.62783i | 1.16272 | + | 7.34115i | −7.81779 | − | 2.54015i | 0 | ||||
18.3 | 1.61837 | + | 0.824603i | −3.42034 | − | 0.541729i | −0.411975 | − | 0.567034i | 0 | −5.08868 | − | 3.69715i | 8.06323 | − | 8.06323i | −1.33571 | − | 8.43332i | 2.84577 | + | 0.924645i | 0 | ||||
18.4 | 2.38234 | + | 1.21387i | 3.57679 | + | 0.566508i | 1.85096 | + | 2.54762i | 0 | 7.83348 | + | 5.69136i | −6.54971 | + | 6.54971i | −0.355933 | − | 2.24727i | 3.91299 | + | 1.27141i | 0 | ||||
32.1 | −1.80600 | − | 0.286042i | 0.665351 | + | 1.30583i | −0.624420 | − | 0.202886i | 0 | −0.828102 | − | 2.54863i | −3.62927 | − | 3.62927i | 7.58652 | + | 3.86553i | 4.02758 | − | 5.54349i | 0 | ||||
32.2 | −0.287585 | − | 0.0455490i | −1.72787 | − | 3.39113i | −3.72360 | − | 1.20987i | 0 | 0.342446 | + | 1.05394i | 2.38950 | + | 2.38950i | 2.05348 | + | 1.04630i | −3.22416 | + | 4.43767i | 0 | ||||
32.3 | 1.86717 | + | 0.295731i | 2.19472 | + | 4.30737i | −0.405347 | − | 0.131705i | 0 | 2.82409 | + | 8.69166i | 3.57009 | + | 3.57009i | −7.45551 | − | 3.79877i | −8.44662 | + | 11.6258i | 0 | ||||
32.4 | 3.57427 | + | 0.566108i | −1.61679 | − | 3.17313i | 8.65068 | + | 2.81078i | 0 | −3.98250 | − | 12.2569i | 0.574149 | + | 0.574149i | 16.4311 | + | 8.37205i | −2.16466 | + | 2.97940i | 0 | ||||
43.1 | −1.80600 | + | 0.286042i | 0.665351 | − | 1.30583i | −0.624420 | + | 0.202886i | 0 | −0.828102 | + | 2.54863i | −3.62927 | + | 3.62927i | 7.58652 | − | 3.86553i | 4.02758 | + | 5.54349i | 0 | ||||
43.2 | −0.287585 | + | 0.0455490i | −1.72787 | + | 3.39113i | −3.72360 | + | 1.20987i | 0 | 0.342446 | − | 1.05394i | 2.38950 | − | 2.38950i | 2.05348 | − | 1.04630i | −3.22416 | − | 4.43767i | 0 | ||||
43.3 | 1.86717 | − | 0.295731i | 2.19472 | − | 4.30737i | −0.405347 | + | 0.131705i | 0 | 2.82409 | − | 8.69166i | 3.57009 | − | 3.57009i | −7.45551 | + | 3.79877i | −8.44662 | − | 11.6258i | 0 | ||||
43.4 | 3.57427 | − | 0.566108i | −1.61679 | + | 3.17313i | 8.65068 | − | 2.81078i | 0 | −3.98250 | + | 12.2569i | 0.574149 | − | 0.574149i | 16.4311 | − | 8.37205i | −2.16466 | − | 2.97940i | 0 | ||||
82.1 | −0.463000 | − | 2.92327i | 0.866921 | + | 0.441718i | −4.52691 | + | 1.47088i | 0 | 0.889877 | − | 2.73876i | −4.44588 | − | 4.44588i | 1.02103 | + | 2.00389i | −4.73363 | − | 6.51528i | 0 | ||||
82.2 | −0.0933465 | − | 0.589367i | −0.210730 | − | 0.107372i | 3.46559 | − | 1.12604i | 0 | −0.0436108 | + | 0.134220i | 7.64532 | + | 7.64532i | −2.07076 | − | 4.06409i | −5.25719 | − | 7.23590i | 0 | ||||
82.3 | 0.312579 | + | 1.97355i | 4.02069 | + | 2.04864i | 0.00704800 | − | 0.00229003i | 0 | −2.78631 | + | 8.57538i | −3.91191 | − | 3.91191i | 3.63528 | + | 7.13464i | 6.67894 | + | 9.19277i | 0 | ||||
82.4 | 0.513943 | + | 3.24491i | −2.81033 | − | 1.43193i | −6.46108 | + | 2.09933i | 0 | 3.20215 | − | 9.85519i | −7.51823 | − | 7.51823i | −4.16668 | − | 8.17758i | 0.557438 | + | 0.767248i | 0 | ||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
25.f | odd | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 125.3.f.c | 32 | |
5.b | even | 2 | 1 | 25.3.f.a | ✓ | 32 | |
5.c | odd | 4 | 1 | 125.3.f.a | 32 | ||
5.c | odd | 4 | 1 | 125.3.f.b | 32 | ||
15.d | odd | 2 | 1 | 225.3.r.a | 32 | ||
20.d | odd | 2 | 1 | 400.3.bg.c | 32 | ||
25.d | even | 5 | 1 | 125.3.f.b | 32 | ||
25.e | even | 10 | 1 | 125.3.f.a | 32 | ||
25.f | odd | 20 | 1 | 25.3.f.a | ✓ | 32 | |
25.f | odd | 20 | 1 | inner | 125.3.f.c | 32 | |
75.l | even | 20 | 1 | 225.3.r.a | 32 | ||
100.l | even | 20 | 1 | 400.3.bg.c | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
25.3.f.a | ✓ | 32 | 5.b | even | 2 | 1 | |
25.3.f.a | ✓ | 32 | 25.f | odd | 20 | 1 | |
125.3.f.a | 32 | 5.c | odd | 4 | 1 | ||
125.3.f.a | 32 | 25.e | even | 10 | 1 | ||
125.3.f.b | 32 | 5.c | odd | 4 | 1 | ||
125.3.f.b | 32 | 25.d | even | 5 | 1 | ||
125.3.f.c | 32 | 1.a | even | 1 | 1 | trivial | |
125.3.f.c | 32 | 25.f | odd | 20 | 1 | inner | |
225.3.r.a | 32 | 15.d | odd | 2 | 1 | ||
225.3.r.a | 32 | 75.l | even | 20 | 1 | ||
400.3.bg.c | 32 | 20.d | odd | 2 | 1 | ||
400.3.bg.c | 32 | 100.l | even | 20 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - 10 T_{2}^{31} + 55 T_{2}^{30} - 220 T_{2}^{29} + 612 T_{2}^{28} - 1380 T_{2}^{27} + \cdots + 12952801 \) acting on \(S_{3}^{\mathrm{new}}(125, [\chi])\).