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 | −3.32707 | + | 1.69523i | 0.541921 | − | 0.0858318i | 5.84445 | − | 8.04419i | 0 | −1.65750 | + | 1.20425i | −1.68463 | − | 1.68463i | −3.47161 | + | 21.9189i | −8.27320 | + | 2.68812i | 0 | ||||
7.2 | −0.776265 | + | 0.395527i | 1.87175 | − | 0.296456i | −1.90500 | + | 2.62200i | 0 | −1.33572 | + | 0.970456i | 5.60844 | + | 5.60844i | 0.986866 | − | 6.23083i | −5.14394 | + | 1.67137i | 0 | ||||
7.3 | −0.508965 | + | 0.259330i | −5.29495 | + | 0.838638i | −2.15935 | + | 2.97209i | 0 | 2.47746 | − | 1.79998i | −1.66138 | − | 1.66138i | 0.685716 | − | 4.32944i | 18.7737 | − | 6.09994i | 0 | ||||
7.4 | 2.54321 | − | 1.29583i | 2.29349 | − | 0.363254i | 2.43759 | − | 3.35505i | 0 | 5.36211 | − | 3.89580i | 3.40272 | + | 3.40272i | 0.0656701 | − | 0.414625i | −3.43135 | + | 1.11491i | 0 | ||||
18.1 | −3.32707 | − | 1.69523i | 0.541921 | + | 0.0858318i | 5.84445 | + | 8.04419i | 0 | −1.65750 | − | 1.20425i | −1.68463 | + | 1.68463i | −3.47161 | − | 21.9189i | −8.27320 | − | 2.68812i | 0 | ||||
18.2 | −0.776265 | − | 0.395527i | 1.87175 | + | 0.296456i | −1.90500 | − | 2.62200i | 0 | −1.33572 | − | 0.970456i | 5.60844 | − | 5.60844i | 0.986866 | + | 6.23083i | −5.14394 | − | 1.67137i | 0 | ||||
18.3 | −0.508965 | − | 0.259330i | −5.29495 | − | 0.838638i | −2.15935 | − | 2.97209i | 0 | 2.47746 | + | 1.79998i | −1.66138 | + | 1.66138i | 0.685716 | + | 4.32944i | 18.7737 | + | 6.09994i | 0 | ||||
18.4 | 2.54321 | + | 1.29583i | 2.29349 | + | 0.363254i | 2.43759 | + | 3.35505i | 0 | 5.36211 | + | 3.89580i | 3.40272 | − | 3.40272i | 0.0656701 | + | 0.414625i | −3.43135 | − | 1.11491i | 0 | ||||
32.1 | −2.92327 | − | 0.463000i | −0.441718 | − | 0.866921i | 4.52691 | + | 1.47088i | 0 | 0.889877 | + | 2.73876i | −4.44588 | − | 4.44588i | −2.00389 | − | 1.02103i | 4.73363 | − | 6.51528i | 0 | ||||
32.2 | −0.589367 | − | 0.0933465i | 0.107372 | + | 0.210730i | −3.46559 | − | 1.12604i | 0 | −0.0436108 | − | 0.134220i | 7.64532 | + | 7.64532i | 4.06409 | + | 2.07076i | 5.25719 | − | 7.23590i | 0 | ||||
32.3 | 1.97355 | + | 0.312579i | −2.04864 | − | 4.02069i | −0.00704800 | − | 0.00229003i | 0 | −2.78631 | − | 8.57538i | −3.91191 | − | 3.91191i | −7.13464 | − | 3.63528i | −6.67894 | + | 9.19277i | 0 | ||||
32.4 | 3.24491 | + | 0.513943i | 1.43193 | + | 2.81033i | 6.46108 | + | 2.09933i | 0 | 3.20215 | + | 9.85519i | −7.51823 | − | 7.51823i | 8.17758 | + | 4.16668i | −0.557438 | + | 0.767248i | 0 | ||||
43.1 | −2.92327 | + | 0.463000i | −0.441718 | + | 0.866921i | 4.52691 | − | 1.47088i | 0 | 0.889877 | − | 2.73876i | −4.44588 | + | 4.44588i | −2.00389 | + | 1.02103i | 4.73363 | + | 6.51528i | 0 | ||||
43.2 | −0.589367 | + | 0.0933465i | 0.107372 | − | 0.210730i | −3.46559 | + | 1.12604i | 0 | −0.0436108 | + | 0.134220i | 7.64532 | − | 7.64532i | 4.06409 | − | 2.07076i | 5.25719 | + | 7.23590i | 0 | ||||
43.3 | 1.97355 | − | 0.312579i | −2.04864 | + | 4.02069i | −0.00704800 | + | 0.00229003i | 0 | −2.78631 | + | 8.57538i | −3.91191 | + | 3.91191i | −7.13464 | + | 3.63528i | −6.67894 | − | 9.19277i | 0 | ||||
43.4 | 3.24491 | − | 0.513943i | 1.43193 | − | 2.81033i | 6.46108 | − | 2.09933i | 0 | 3.20215 | − | 9.85519i | −7.51823 | + | 7.51823i | 8.17758 | − | 4.16668i | −0.557438 | − | 0.767248i | 0 | ||||
82.1 | −0.286042 | − | 1.80600i | −1.30583 | − | 0.665351i | 0.624420 | − | 0.202886i | 0 | −0.828102 | + | 2.54863i | −3.62927 | − | 3.62927i | −3.86553 | − | 7.58652i | −4.02758 | − | 5.54349i | 0 | ||||
82.2 | −0.0455490 | − | 0.287585i | 3.39113 | + | 1.72787i | 3.72360 | − | 1.20987i | 0 | 0.342446 | − | 1.05394i | 2.38950 | + | 2.38950i | −1.04630 | − | 2.05348i | 3.22416 | + | 4.43767i | 0 | ||||
82.3 | 0.295731 | + | 1.86717i | −4.30737 | − | 2.19472i | 0.405347 | − | 0.131705i | 0 | 2.82409 | − | 8.69166i | 3.57009 | + | 3.57009i | 3.79877 | + | 7.45551i | 8.44662 | + | 11.6258i | 0 | ||||
82.4 | 0.566108 | + | 3.57427i | 3.17313 | + | 1.61679i | −8.65068 | + | 2.81078i | 0 | −3.98250 | + | 12.2569i | 0.574149 | + | 0.574149i | −8.37205 | − | 16.4311i | 2.16466 | + | 2.97940i | 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.b | 32 | |
5.b | even | 2 | 1 | 125.3.f.a | 32 | ||
5.c | odd | 4 | 1 | 25.3.f.a | ✓ | 32 | |
5.c | odd | 4 | 1 | 125.3.f.c | 32 | ||
15.e | even | 4 | 1 | 225.3.r.a | 32 | ||
20.e | even | 4 | 1 | 400.3.bg.c | 32 | ||
25.d | even | 5 | 1 | 125.3.f.c | 32 | ||
25.e | even | 10 | 1 | 25.3.f.a | ✓ | 32 | |
25.f | odd | 20 | 1 | 125.3.f.a | 32 | ||
25.f | odd | 20 | 1 | inner | 125.3.f.b | 32 | |
75.h | odd | 10 | 1 | 225.3.r.a | 32 | ||
100.h | odd | 10 | 1 | 400.3.bg.c | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
25.3.f.a | ✓ | 32 | 5.c | odd | 4 | 1 | |
25.3.f.a | ✓ | 32 | 25.e | even | 10 | 1 | |
125.3.f.a | 32 | 5.b | even | 2 | 1 | ||
125.3.f.a | 32 | 25.f | odd | 20 | 1 | ||
125.3.f.b | 32 | 1.a | even | 1 | 1 | trivial | |
125.3.f.b | 32 | 25.f | odd | 20 | 1 | inner | |
125.3.f.c | 32 | 5.c | odd | 4 | 1 | ||
125.3.f.c | 32 | 25.d | even | 5 | 1 | ||
225.3.r.a | 32 | 15.e | even | 4 | 1 | ||
225.3.r.a | 32 | 75.h | odd | 10 | 1 | ||
400.3.bg.c | 32 | 20.e | even | 4 | 1 | ||
400.3.bg.c | 32 | 100.h | odd | 10 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - 5 T_{2}^{30} + 10 T_{2}^{29} - 88 T_{2}^{28} - 460 T_{2}^{27} + 1255 T_{2}^{26} + \cdots + 12952801 \) acting on \(S_{3}^{\mathrm{new}}(125, [\chi])\).