Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [600,2,Mod(17,600)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(600, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([0, 0, 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("600.17");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 600.bu (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: | \(4.79102412128\) |
Analytic rank: | \(0\) |
Dimension: | \(240\) |
Relative dimension: | \(30\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 | 0 | −1.68774 | + | 0.389291i | 0 | −1.62862 | − | 1.53218i | 0 | −0.785661 | − | 0.785661i | 0 | 2.69691 | − | 1.31404i | 0 | ||||||||||
17.2 | 0 | −1.65974 | + | 0.495252i | 0 | 1.48459 | + | 1.67212i | 0 | 3.63505 | + | 3.63505i | 0 | 2.50945 | − | 1.64398i | 0 | ||||||||||
17.3 | 0 | −1.62090 | − | 0.610467i | 0 | 2.16992 | − | 0.539844i | 0 | 0.548883 | + | 0.548883i | 0 | 2.25466 | + | 1.97902i | 0 | ||||||||||
17.4 | 0 | −1.59112 | − | 0.684344i | 0 | −1.02804 | + | 1.98573i | 0 | −1.26254 | − | 1.26254i | 0 | 2.06335 | + | 2.17775i | 0 | ||||||||||
17.5 | 0 | −1.56705 | − | 0.737803i | 0 | −2.23532 | − | 0.0579740i | 0 | 1.32422 | + | 1.32422i | 0 | 1.91129 | + | 2.31235i | 0 | ||||||||||
17.6 | 0 | −1.48940 | + | 0.884135i | 0 | −1.32812 | + | 1.79892i | 0 | 1.66646 | + | 1.66646i | 0 | 1.43661 | − | 2.63366i | 0 | ||||||||||
17.7 | 0 | −1.43901 | + | 0.963981i | 0 | 1.87829 | + | 1.21326i | 0 | −2.44789 | − | 2.44789i | 0 | 1.14148 | − | 2.77435i | 0 | ||||||||||
17.8 | 0 | −1.24488 | + | 1.20427i | 0 | 0.125563 | − | 2.23254i | 0 | 0.631710 | + | 0.631710i | 0 | 0.0994587 | − | 2.99835i | 0 | ||||||||||
17.9 | 0 | −1.16124 | − | 1.28512i | 0 | 1.95784 | − | 1.08022i | 0 | −2.10037 | − | 2.10037i | 0 | −0.303061 | + | 2.98465i | 0 | ||||||||||
17.10 | 0 | −1.04119 | − | 1.38417i | 0 | −0.137473 | − | 2.23184i | 0 | 2.98357 | + | 2.98357i | 0 | −0.831844 | + | 2.88237i | 0 | ||||||||||
17.11 | 0 | −0.705878 | + | 1.58169i | 0 | 1.35766 | − | 1.77673i | 0 | 0.131544 | + | 0.131544i | 0 | −2.00347 | − | 2.23296i | 0 | ||||||||||
17.12 | 0 | −0.619322 | − | 1.61754i | 0 | −0.132259 | + | 2.23215i | 0 | −3.14250 | − | 3.14250i | 0 | −2.23288 | + | 2.00356i | 0 | ||||||||||
17.13 | 0 | −0.450732 | + | 1.67238i | 0 | 1.67263 | + | 1.48402i | 0 | −2.25950 | − | 2.25950i | 0 | −2.59368 | − | 1.50759i | 0 | ||||||||||
17.14 | 0 | −0.128084 | − | 1.72731i | 0 | −1.29140 | − | 1.82546i | 0 | −0.526415 | − | 0.526415i | 0 | −2.96719 | + | 0.442481i | 0 | ||||||||||
17.15 | 0 | −0.0881207 | + | 1.72981i | 0 | −1.67263 | − | 1.48402i | 0 | −2.25950 | − | 2.25950i | 0 | −2.98447 | − | 0.304864i | 0 | ||||||||||
17.16 | 0 | 0.0788384 | − | 1.73026i | 0 | −2.06252 | + | 0.863731i | 0 | 1.38221 | + | 1.38221i | 0 | −2.98757 | − | 0.272821i | 0 | ||||||||||
17.17 | 0 | 0.182562 | + | 1.72240i | 0 | −1.35766 | + | 1.77673i | 0 | 0.131544 | + | 0.131544i | 0 | −2.93334 | + | 0.628890i | 0 | ||||||||||
17.18 | 0 | 0.459699 | − | 1.66993i | 0 | 2.06252 | − | 0.863731i | 0 | 1.38221 | + | 1.38221i | 0 | −2.57735 | − | 1.53533i | 0 | ||||||||||
17.19 | 0 | 0.655583 | − | 1.60319i | 0 | 1.29140 | + | 1.82546i | 0 | −0.526415 | − | 0.526415i | 0 | −2.14042 | − | 2.10204i | 0 | ||||||||||
17.20 | 0 | 0.811812 | + | 1.53002i | 0 | −0.125563 | + | 2.23254i | 0 | 0.631710 | + | 0.631710i | 0 | −1.68192 | + | 2.48418i | 0 | ||||||||||
See next 80 embeddings (of 240 total) |
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 | 600.2.bu.a | ✓ | 240 |
3.b | odd | 2 | 1 | inner | 600.2.bu.a | ✓ | 240 |
25.f | odd | 20 | 1 | inner | 600.2.bu.a | ✓ | 240 |
75.l | even | 20 | 1 | inner | 600.2.bu.a | ✓ | 240 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
600.2.bu.a | ✓ | 240 | 1.a | even | 1 | 1 | trivial |
600.2.bu.a | ✓ | 240 | 3.b | odd | 2 | 1 | inner |
600.2.bu.a | ✓ | 240 | 25.f | odd | 20 | 1 | inner |
600.2.bu.a | ✓ | 240 | 75.l | even | 20 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(600, [\chi])\).