Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [570,2,Mod(139,570)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(570, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([0, 9, 14]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("570.139");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 570.bc (of order \(18\), degree \(6\), 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: | \(4.55147291521\) |
Analytic rank: | \(0\) |
Dimension: | \(60\) |
Relative dimension: | \(10\) over \(\Q(\zeta_{18})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{18}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
139.1 | −0.642788 | − | 0.766044i | −0.342020 | + | 0.939693i | −0.173648 | + | 0.984808i | −2.14038 | + | 0.647126i | 0.939693 | − | 0.342020i | −0.536979 | − | 0.310025i | 0.866025 | − | 0.500000i | −0.766044 | − | 0.642788i | 1.87154 | + | 1.22366i |
139.2 | −0.642788 | − | 0.766044i | −0.342020 | + | 0.939693i | −0.173648 | + | 0.984808i | −1.78880 | − | 1.34171i | 0.939693 | − | 0.342020i | 3.95160 | + | 2.28146i | 0.866025 | − | 0.500000i | −0.766044 | − | 0.642788i | 0.122006 | + | 2.23274i |
139.3 | −0.642788 | − | 0.766044i | −0.342020 | + | 0.939693i | −0.173648 | + | 0.984808i | −1.32288 | − | 1.80277i | 0.939693 | − | 0.342020i | −3.17019 | − | 1.83031i | 0.866025 | − | 0.500000i | −0.766044 | − | 0.642788i | −0.530667 | + | 2.17219i |
139.4 | −0.642788 | − | 0.766044i | −0.342020 | + | 0.939693i | −0.173648 | + | 0.984808i | 2.18336 | − | 0.482631i | 0.939693 | − | 0.342020i | −2.55062 | − | 1.47260i | 0.866025 | − | 0.500000i | −0.766044 | − | 0.642788i | −1.77315 | − | 1.36232i |
139.5 | −0.642788 | − | 0.766044i | −0.342020 | + | 0.939693i | −0.173648 | + | 0.984808i | 2.22668 | + | 0.204655i | 0.939693 | − | 0.342020i | 2.92703 | + | 1.68992i | 0.866025 | − | 0.500000i | −0.766044 | − | 0.642788i | −1.27451 | − | 1.83729i |
139.6 | 0.642788 | + | 0.766044i | 0.342020 | − | 0.939693i | −0.173648 | + | 0.984808i | −2.21676 | + | 0.293229i | 0.939693 | − | 0.342020i | 2.55062 | + | 1.47260i | −0.866025 | + | 0.500000i | −0.766044 | − | 0.642788i | −1.64953 | − | 1.50965i |
139.7 | 0.642788 | + | 0.766044i | 0.342020 | − | 0.939693i | −0.173648 | + | 0.984808i | −2.02240 | + | 0.953883i | 0.939693 | − | 0.342020i | −2.92703 | − | 1.68992i | −0.866025 | + | 0.500000i | −0.766044 | − | 0.642788i | −2.03069 | − | 0.936105i |
139.8 | 0.642788 | + | 0.766044i | 0.342020 | − | 0.939693i | −0.173648 | + | 0.984808i | 0.626522 | − | 2.14650i | 0.939693 | − | 0.342020i | 3.17019 | + | 1.83031i | −0.866025 | + | 0.500000i | −0.766044 | − | 0.642788i | 2.04704 | − | 0.899801i |
139.9 | 0.642788 | + | 0.766044i | 0.342020 | − | 0.939693i | −0.173648 | + | 0.984808i | 1.22203 | − | 1.87260i | 0.939693 | − | 0.342020i | −3.95160 | − | 2.28146i | −0.866025 | + | 0.500000i | −0.766044 | − | 0.642788i | 2.22000 | − | 0.267559i |
139.10 | 0.642788 | + | 0.766044i | 0.342020 | − | 0.939693i | −0.173648 | + | 0.984808i | 2.23263 | − | 0.123954i | 0.939693 | − | 0.342020i | 0.536979 | + | 0.310025i | −0.866025 | + | 0.500000i | −0.766044 | − | 0.642788i | 1.53006 | + | 1.63062i |
169.1 | −0.984808 | − | 0.173648i | 0.642788 | − | 0.766044i | 0.939693 | + | 0.342020i | −2.14282 | − | 0.639010i | −0.766044 | + | 0.642788i | −0.495802 | + | 0.286252i | −0.866025 | − | 0.500000i | −0.173648 | − | 0.984808i | 1.99930 | + | 1.00140i |
169.2 | −0.984808 | − | 0.173648i | 0.642788 | − | 0.766044i | 0.939693 | + | 0.342020i | −0.318910 | + | 2.21321i | −0.766044 | + | 0.642788i | −2.49117 | + | 1.43828i | −0.866025 | − | 0.500000i | −0.173648 | − | 0.984808i | 0.698385 | − | 2.12421i |
169.3 | −0.984808 | − | 0.173648i | 0.642788 | − | 0.766044i | 0.939693 | + | 0.342020i | −0.174899 | − | 2.22922i | −0.766044 | + | 0.642788i | −2.19817 | + | 1.26911i | −0.866025 | − | 0.500000i | −0.173648 | − | 0.984808i | −0.214857 | + | 2.22572i |
169.4 | −0.984808 | − | 0.173648i | 0.642788 | − | 0.766044i | 0.939693 | + | 0.342020i | 0.588223 | + | 2.15731i | −0.766044 | + | 0.642788i | 4.50054 | − | 2.59839i | −0.866025 | − | 0.500000i | −0.173648 | − | 0.984808i | −0.204673 | − | 2.22668i |
169.5 | −0.984808 | − | 0.173648i | 0.642788 | − | 0.766044i | 0.939693 | + | 0.342020i | 2.19119 | + | 0.445735i | −0.766044 | + | 0.642788i | 0.187519 | − | 0.108264i | −0.866025 | − | 0.500000i | −0.173648 | − | 0.984808i | −2.08050 | − | 0.819460i |
169.6 | 0.984808 | + | 0.173648i | −0.642788 | + | 0.766044i | 0.939693 | + | 0.342020i | −1.66692 | − | 1.49043i | −0.766044 | + | 0.642788i | 2.49117 | − | 1.43828i | 0.866025 | + | 0.500000i | −0.173648 | − | 0.984808i | −1.38279 | − | 1.75724i |
169.7 | 0.984808 | + | 0.173648i | −0.642788 | + | 0.766044i | 0.939693 | + | 0.342020i | −1.23075 | + | 1.86689i | −0.766044 | + | 0.642788i | 0.495802 | − | 0.286252i | 0.866025 | + | 0.500000i | −0.173648 | − | 0.984808i | −1.53623 | + | 1.62481i |
169.8 | 0.984808 | + | 0.173648i | −0.642788 | + | 0.766044i | 0.939693 | + | 0.342020i | −0.936089 | − | 2.03070i | −0.766044 | + | 0.642788i | −4.50054 | + | 2.59839i | 0.866025 | + | 0.500000i | −0.173648 | − | 0.984808i | −0.569240 | − | 2.16240i |
169.9 | 0.984808 | + | 0.173648i | −0.642788 | + | 0.766044i | 0.939693 | + | 0.342020i | 1.29893 | + | 1.82010i | −0.766044 | + | 0.642788i | 2.19817 | − | 1.26911i | 0.866025 | + | 0.500000i | −0.173648 | − | 0.984808i | 0.963141 | + | 2.01801i |
169.10 | 0.984808 | + | 0.173648i | −0.642788 | + | 0.766044i | 0.939693 | + | 0.342020i | 1.39204 | − | 1.74992i | −0.766044 | + | 0.642788i | −0.187519 | + | 0.108264i | 0.866025 | + | 0.500000i | −0.173648 | − | 0.984808i | 1.67476 | − | 1.48161i |
See all 60 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
19.e | even | 9 | 1 | inner |
95.p | even | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 570.2.bc.a | ✓ | 60 |
5.b | even | 2 | 1 | inner | 570.2.bc.a | ✓ | 60 |
19.e | even | 9 | 1 | inner | 570.2.bc.a | ✓ | 60 |
95.p | even | 18 | 1 | inner | 570.2.bc.a | ✓ | 60 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
570.2.bc.a | ✓ | 60 | 1.a | even | 1 | 1 | trivial |
570.2.bc.a | ✓ | 60 | 5.b | even | 2 | 1 | inner |
570.2.bc.a | ✓ | 60 | 19.e | even | 9 | 1 | inner |
570.2.bc.a | ✓ | 60 | 95.p | even | 18 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{60} - 138 T_{7}^{58} + 10737 T_{7}^{56} - 571100 T_{7}^{54} + 22951392 T_{7}^{52} + \cdots + 50\!\cdots\!21 \) acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\).