Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [25,3,Mod(2,25)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(25, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("25.2");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 25 = 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 25.f (of order \(20\), degree \(8\), 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: | \(0.681200660901\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{20})\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | −3.57427 | − | 0.566108i | 1.61679 | + | 3.17313i | 8.65068 | + | 2.81078i | −0.872190 | + | 4.92334i | −3.98250 | − | 12.2569i | −0.574149 | − | 0.574149i | −16.4311 | − | 8.37205i | −2.16466 | + | 2.97940i | 5.90458 | − | 17.1036i |
2.2 | −1.86717 | − | 0.295731i | −2.19472 | − | 4.30737i | −0.405347 | − | 0.131705i | 4.99561 | + | 0.209511i | 2.82409 | + | 8.69166i | −3.57009 | − | 3.57009i | 7.45551 | + | 3.79877i | −8.44662 | + | 11.6258i | −9.26571 | − | 1.86855i |
2.3 | 0.287585 | + | 0.0455490i | 1.72787 | + | 3.39113i | −3.72360 | − | 1.20987i | 2.36408 | − | 4.40581i | 0.342446 | + | 1.05394i | −2.38950 | − | 2.38950i | −2.05348 | − | 1.04630i | −3.22416 | + | 4.43767i | 0.880552 | − | 1.15936i |
2.4 | 1.80600 | + | 0.286042i | −0.665351 | − | 1.30583i | −0.624420 | − | 0.202886i | −3.20727 | + | 3.83580i | −0.828102 | − | 2.54863i | 3.62927 | + | 3.62927i | −7.58652 | − | 3.86553i | 4.02758 | − | 5.54349i | −6.88953 | + | 6.01004i |
3.1 | −1.69523 | − | 3.32707i | −0.0858318 | − | 0.541921i | −5.84445 | + | 8.04419i | 2.26962 | − | 4.45520i | −1.65750 | + | 1.20425i | 1.68463 | − | 1.68463i | 21.9189 | + | 3.47161i | 8.27320 | − | 2.68812i | −18.6703 | + | 0.00137996i |
3.2 | −0.395527 | − | 0.776265i | −0.296456 | − | 1.87175i | 1.90500 | − | 2.62200i | 1.22928 | + | 4.84653i | −1.33572 | + | 0.970456i | −5.60844 | + | 5.60844i | −6.23083 | − | 0.986866i | 5.14394 | − | 1.67137i | 3.27598 | − | 2.87118i |
3.3 | −0.259330 | − | 0.508965i | 0.838638 | + | 5.29495i | 2.15935 | − | 2.97209i | −3.73307 | − | 3.32629i | 2.47746 | − | 1.79998i | 1.66138 | − | 1.66138i | −4.32944 | − | 0.685716i | −18.7737 | + | 6.09994i | −0.724866 | + | 2.76261i |
3.4 | 1.29583 | + | 2.54321i | −0.363254 | − | 2.29349i | −2.43759 | + | 3.35505i | −4.45624 | − | 2.26758i | 5.36211 | − | 3.89580i | −3.40272 | + | 3.40272i | −0.414625 | − | 0.0656701i | 3.43135 | − | 1.11491i | −0.00760491 | − | 14.2715i |
8.1 | −2.38234 | − | 1.21387i | −3.57679 | − | 0.566508i | 1.85096 | + | 2.54762i | −4.45026 | − | 2.27929i | 7.83348 | + | 5.69136i | 6.54971 | − | 6.54971i | 0.355933 | + | 2.24727i | 3.91299 | + | 1.27141i | 7.83532 | + | 10.8321i |
8.2 | −1.61837 | − | 0.824603i | 3.42034 | + | 0.541729i | −0.411975 | − | 0.567034i | 4.00059 | − | 2.99921i | −5.08868 | − | 3.69715i | −8.06323 | + | 8.06323i | 1.33571 | + | 8.43332i | 2.84577 | + | 0.924645i | −8.94762 | + | 1.55494i |
8.3 | 0.972743 | + | 0.495637i | 0.872241 | + | 0.138149i | −1.65057 | − | 2.27181i | −2.66494 | + | 4.23062i | 0.779995 | + | 0.566699i | 1.62783 | − | 1.62783i | −1.16272 | − | 7.34115i | −7.81779 | − | 2.54015i | −4.68915 | + | 2.79446i |
8.4 | 2.70026 | + | 1.37585i | −4.42692 | − | 0.701156i | 3.04732 | + | 4.19427i | 1.95091 | − | 4.60369i | −10.9892 | − | 7.98410i | −4.77540 | + | 4.77540i | 0.561510 | + | 3.54523i | 10.5465 | + | 3.42677i | 11.6020 | − | 9.74701i |
12.1 | −0.513943 | − | 3.24491i | 2.81033 | + | 1.43193i | −6.46108 | + | 2.09933i | −4.99960 | − | 0.0628765i | 3.20215 | − | 9.85519i | 7.51823 | + | 7.51823i | 4.16668 | + | 8.17758i | 0.557438 | + | 0.767248i | 2.36548 | + | 16.2556i |
12.2 | −0.312579 | − | 1.97355i | −4.02069 | − | 2.04864i | 0.00704800 | − | 0.00229003i | 4.93389 | + | 0.810386i | −2.78631 | + | 8.57538i | 3.91191 | + | 3.91191i | −3.63528 | − | 7.13464i | 6.67894 | + | 9.19277i | 0.0571038 | − | 9.99057i |
12.3 | 0.0933465 | + | 0.589367i | 0.210730 | + | 0.107372i | 3.46559 | − | 1.12604i | −3.31432 | + | 3.74370i | −0.0436108 | + | 0.134220i | −7.64532 | − | 7.64532i | 2.07076 | + | 4.06409i | −5.25719 | − | 7.23590i | −2.51579 | − | 1.60389i |
12.4 | 0.463000 | + | 2.92327i | −0.866921 | − | 0.441718i | −4.52691 | + | 1.47088i | 0.953911 | − | 4.90816i | 0.889877 | − | 2.73876i | 4.44588 | + | 4.44588i | −1.02103 | − | 2.00389i | −4.73363 | − | 6.51528i | 14.7895 | + | 0.516059i |
13.1 | −3.57427 | + | 0.566108i | 1.61679 | − | 3.17313i | 8.65068 | − | 2.81078i | −0.872190 | − | 4.92334i | −3.98250 | + | 12.2569i | −0.574149 | + | 0.574149i | −16.4311 | + | 8.37205i | −2.16466 | − | 2.97940i | 5.90458 | + | 17.1036i |
13.2 | −1.86717 | + | 0.295731i | −2.19472 | + | 4.30737i | −0.405347 | + | 0.131705i | 4.99561 | − | 0.209511i | 2.82409 | − | 8.69166i | −3.57009 | + | 3.57009i | 7.45551 | − | 3.79877i | −8.44662 | − | 11.6258i | −9.26571 | + | 1.86855i |
13.3 | 0.287585 | − | 0.0455490i | 1.72787 | − | 3.39113i | −3.72360 | + | 1.20987i | 2.36408 | + | 4.40581i | 0.342446 | − | 1.05394i | −2.38950 | + | 2.38950i | −2.05348 | + | 1.04630i | −3.22416 | − | 4.43767i | 0.880552 | + | 1.15936i |
13.4 | 1.80600 | − | 0.286042i | −0.665351 | + | 1.30583i | −0.624420 | + | 0.202886i | −3.20727 | − | 3.83580i | −0.828102 | + | 2.54863i | 3.62927 | − | 3.62927i | −7.58652 | + | 3.86553i | 4.02758 | + | 5.54349i | −6.88953 | − | 6.01004i |
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 | 25.3.f.a | ✓ | 32 |
3.b | odd | 2 | 1 | 225.3.r.a | 32 | ||
4.b | odd | 2 | 1 | 400.3.bg.c | 32 | ||
5.b | even | 2 | 1 | 125.3.f.c | 32 | ||
5.c | odd | 4 | 1 | 125.3.f.a | 32 | ||
5.c | odd | 4 | 1 | 125.3.f.b | 32 | ||
25.d | even | 5 | 1 | 125.3.f.a | 32 | ||
25.e | even | 10 | 1 | 125.3.f.b | 32 | ||
25.f | odd | 20 | 1 | inner | 25.3.f.a | ✓ | 32 |
25.f | odd | 20 | 1 | 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 | 1.a | even | 1 | 1 | trivial |
25.3.f.a | ✓ | 32 | 25.f | odd | 20 | 1 | inner |
125.3.f.a | 32 | 5.c | odd | 4 | 1 | ||
125.3.f.a | 32 | 25.d | even | 5 | 1 | ||
125.3.f.b | 32 | 5.c | odd | 4 | 1 | ||
125.3.f.b | 32 | 25.e | even | 10 | 1 | ||
125.3.f.c | 32 | 5.b | even | 2 | 1 | ||
125.3.f.c | 32 | 25.f | odd | 20 | 1 | ||
225.3.r.a | 32 | 3.b | odd | 2 | 1 | ||
225.3.r.a | 32 | 75.l | even | 20 | 1 | ||
400.3.bg.c | 32 | 4.b | odd | 2 | 1 | ||
400.3.bg.c | 32 | 100.l | even | 20 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(25, [\chi])\).