Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [222,2,Mod(25,222)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(222, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([0, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("222.25");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 222 = 2 \cdot 3 \cdot 37 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 222.n (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: | \(1.77267892487\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(4\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
25.1 | −0.642788 | − | 0.766044i | 0.766044 | + | 0.642788i | −0.173648 | + | 0.984808i | −1.39618 | − | 3.83596i | − | 1.00000i | 1.12643 | − | 0.409987i | 0.866025 | − | 0.500000i | 0.173648 | + | 0.984808i | −2.04107 | + | 3.53524i | |
25.2 | −0.642788 | − | 0.766044i | 0.766044 | + | 0.642788i | −0.173648 | + | 0.984808i | 0.753389 | + | 2.06992i | − | 1.00000i | −2.61124 | + | 0.950413i | 0.866025 | − | 0.500000i | 0.173648 | + | 0.984808i | 1.10138 | − | 1.90765i | |
25.3 | 0.642788 | + | 0.766044i | 0.766044 | + | 0.642788i | −0.173648 | + | 0.984808i | −0.100476 | − | 0.276055i | 1.00000i | 2.48902 | − | 0.905929i | −0.866025 | + | 0.500000i | 0.173648 | + | 0.984808i | 0.146886 | − | 0.254414i | ||
25.4 | 0.642788 | + | 0.766044i | 0.766044 | + | 0.642788i | −0.173648 | + | 0.984808i | 0.743263 | + | 2.04210i | 1.00000i | −2.00421 | + | 0.729473i | −0.866025 | + | 0.500000i | 0.173648 | + | 0.984808i | −1.08658 | + | 1.88201i | ||
67.1 | −0.342020 | − | 0.939693i | −0.939693 | − | 0.342020i | −0.766044 | + | 0.642788i | −2.10406 | + | 0.371002i | 1.00000i | 0.485910 | + | 2.75573i | 0.866025 | + | 0.500000i | 0.766044 | + | 0.642788i | 1.06826 | + | 1.85028i | ||
67.2 | −0.342020 | − | 0.939693i | −0.939693 | − | 0.342020i | −0.766044 | + | 0.642788i | 1.76204 | − | 0.310695i | 1.00000i | −0.343122 | − | 1.94594i | 0.866025 | + | 0.500000i | 0.766044 | + | 0.642788i | −0.894610 | − | 1.54951i | ||
67.3 | 0.342020 | + | 0.939693i | −0.939693 | − | 0.342020i | −0.766044 | + | 0.642788i | −3.64700 | + | 0.643064i | − | 1.00000i | −0.469097 | − | 2.66038i | −0.866025 | − | 0.500000i | 0.766044 | + | 0.642788i | −1.85163 | − | 3.20712i | |
67.4 | 0.342020 | + | 0.939693i | −0.939693 | − | 0.342020i | −0.766044 | + | 0.642788i | 3.98902 | − | 0.703371i | − | 1.00000i | −0.673690 | − | 3.82069i | −0.866025 | − | 0.500000i | 0.766044 | + | 0.642788i | 2.02528 | + | 3.50788i | |
115.1 | −0.984808 | − | 0.173648i | 0.173648 | + | 0.984808i | 0.939693 | + | 0.342020i | −2.32681 | − | 2.77298i | − | 1.00000i | −2.40991 | + | 2.02215i | −0.866025 | − | 0.500000i | −0.939693 | + | 0.342020i | 1.80993 | + | 3.13490i | |
115.2 | −0.984808 | − | 0.173648i | 0.173648 | + | 0.984808i | 0.939693 | + | 0.342020i | 1.34200 | + | 1.59933i | − | 1.00000i | 1.56789 | − | 1.31561i | −0.866025 | − | 0.500000i | −0.939693 | + | 0.342020i | −1.04389 | − | 1.80807i | |
115.3 | 0.984808 | + | 0.173648i | 0.173648 | + | 0.984808i | 0.939693 | + | 0.342020i | 0.00457474 | + | 0.00545196i | 1.00000i | 2.74008 | − | 2.29920i | 0.866025 | + | 0.500000i | −0.939693 | + | 0.342020i | 0.00355851 | + | 0.00616353i | ||
115.4 | 0.984808 | + | 0.173648i | 0.173648 | + | 0.984808i | 0.939693 | + | 0.342020i | 0.980233 | + | 1.16820i | 1.00000i | −2.89806 | + | 2.43176i | 0.866025 | + | 0.500000i | −0.939693 | + | 0.342020i | 0.762486 | + | 1.32066i | ||
139.1 | −0.984808 | + | 0.173648i | 0.173648 | − | 0.984808i | 0.939693 | − | 0.342020i | −2.32681 | + | 2.77298i | 1.00000i | −2.40991 | − | 2.02215i | −0.866025 | + | 0.500000i | −0.939693 | − | 0.342020i | 1.80993 | − | 3.13490i | ||
139.2 | −0.984808 | + | 0.173648i | 0.173648 | − | 0.984808i | 0.939693 | − | 0.342020i | 1.34200 | − | 1.59933i | 1.00000i | 1.56789 | + | 1.31561i | −0.866025 | + | 0.500000i | −0.939693 | − | 0.342020i | −1.04389 | + | 1.80807i | ||
139.3 | 0.984808 | − | 0.173648i | 0.173648 | − | 0.984808i | 0.939693 | − | 0.342020i | 0.00457474 | − | 0.00545196i | − | 1.00000i | 2.74008 | + | 2.29920i | 0.866025 | − | 0.500000i | −0.939693 | − | 0.342020i | 0.00355851 | − | 0.00616353i | |
139.4 | 0.984808 | − | 0.173648i | 0.173648 | − | 0.984808i | 0.939693 | − | 0.342020i | 0.980233 | − | 1.16820i | − | 1.00000i | −2.89806 | − | 2.43176i | 0.866025 | − | 0.500000i | −0.939693 | − | 0.342020i | 0.762486 | − | 1.32066i | |
151.1 | −0.642788 | + | 0.766044i | 0.766044 | − | 0.642788i | −0.173648 | − | 0.984808i | −1.39618 | + | 3.83596i | 1.00000i | 1.12643 | + | 0.409987i | 0.866025 | + | 0.500000i | 0.173648 | − | 0.984808i | −2.04107 | − | 3.53524i | ||
151.2 | −0.642788 | + | 0.766044i | 0.766044 | − | 0.642788i | −0.173648 | − | 0.984808i | 0.753389 | − | 2.06992i | 1.00000i | −2.61124 | − | 0.950413i | 0.866025 | + | 0.500000i | 0.173648 | − | 0.984808i | 1.10138 | + | 1.90765i | ||
151.3 | 0.642788 | − | 0.766044i | 0.766044 | − | 0.642788i | −0.173648 | − | 0.984808i | −0.100476 | + | 0.276055i | − | 1.00000i | 2.48902 | + | 0.905929i | −0.866025 | − | 0.500000i | 0.173648 | − | 0.984808i | 0.146886 | + | 0.254414i | |
151.4 | 0.642788 | − | 0.766044i | 0.766044 | − | 0.642788i | −0.173648 | − | 0.984808i | 0.743263 | − | 2.04210i | − | 1.00000i | −2.00421 | − | 0.729473i | −0.866025 | − | 0.500000i | 0.173648 | − | 0.984808i | −1.08658 | − | 1.88201i | |
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
37.h | even | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 222.2.n.a | ✓ | 24 |
3.b | odd | 2 | 1 | 666.2.bj.e | 24 | ||
37.h | even | 18 | 1 | inner | 222.2.n.a | ✓ | 24 |
37.i | odd | 36 | 1 | 8214.2.a.bh | 12 | ||
37.i | odd | 36 | 1 | 8214.2.a.bj | 12 | ||
111.n | odd | 18 | 1 | 666.2.bj.e | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
222.2.n.a | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
222.2.n.a | ✓ | 24 | 37.h | even | 18 | 1 | inner |
666.2.bj.e | 24 | 3.b | odd | 2 | 1 | ||
666.2.bj.e | 24 | 111.n | odd | 18 | 1 | ||
8214.2.a.bh | 12 | 37.i | odd | 36 | 1 | ||
8214.2.a.bj | 12 | 37.i | odd | 36 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - 12 T_{5}^{22} - 54 T_{5}^{21} - 51 T_{5}^{20} + 882 T_{5}^{19} + 260 T_{5}^{18} + \cdots + 729 \) acting on \(S_{2}^{\mathrm{new}}(222, [\chi])\).