Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [456,2,Mod(41,456)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(456, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([0, 0, 9, 13]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("456.41");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 456 = 2^{3} \cdot 3 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 456.bm (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: | \(3.64117833217\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
41.1 | 0 | −1.55056 | + | 0.771852i | 0 | −3.00241 | − | 0.529406i | 0 | −0.251171 | − | 0.435040i | 0 | 1.80849 | − | 2.39361i | 0 | ||||||||||
41.2 | 0 | −1.50282 | − | 0.861118i | 0 | −0.320800 | − | 0.0565656i | 0 | −0.0144179 | − | 0.0249726i | 0 | 1.51695 | + | 2.58822i | 0 | ||||||||||
41.3 | 0 | −1.35104 | + | 1.08383i | 0 | 3.33482 | + | 0.588019i | 0 | 2.19892 | + | 3.80864i | 0 | 0.650612 | − | 2.92860i | 0 | ||||||||||
41.4 | 0 | −0.0815079 | + | 1.73013i | 0 | −1.23805 | − | 0.218301i | 0 | −0.753669 | − | 1.30539i | 0 | −2.98671 | − | 0.282039i | 0 | ||||||||||
41.5 | 0 | 0.144474 | − | 1.72601i | 0 | 2.45783 | + | 0.433381i | 0 | −2.29577 | − | 3.97638i | 0 | −2.95825 | − | 0.498730i | 0 | ||||||||||
41.6 | 0 | 0.264929 | − | 1.71167i | 0 | −4.07516 | − | 0.718561i | 0 | 0.904376 | + | 1.56643i | 0 | −2.85963 | − | 0.906941i | 0 | ||||||||||
41.7 | 0 | 0.505407 | + | 1.65667i | 0 | 2.78114 | + | 0.490390i | 0 | −1.12097 | − | 1.94159i | 0 | −2.48913 | + | 1.67459i | 0 | ||||||||||
41.8 | 0 | 1.20462 | − | 1.24454i | 0 | 1.70062 | + | 0.299865i | 0 | 1.13747 | + | 1.97015i | 0 | −0.0977739 | − | 2.99841i | 0 | ||||||||||
41.9 | 0 | 1.57581 | + | 0.718900i | 0 | 0.726176 | + | 0.128044i | 0 | 1.38478 | + | 2.39851i | 0 | 1.96636 | + | 2.26570i | 0 | ||||||||||
41.10 | 0 | 1.73038 | − | 0.0760235i | 0 | −2.36417 | − | 0.416867i | 0 | −2.12924 | − | 3.68795i | 0 | 2.98844 | − | 0.263099i | 0 | ||||||||||
89.1 | 0 | −1.55056 | − | 0.771852i | 0 | −3.00241 | + | 0.529406i | 0 | −0.251171 | + | 0.435040i | 0 | 1.80849 | + | 2.39361i | 0 | ||||||||||
89.2 | 0 | −1.50282 | + | 0.861118i | 0 | −0.320800 | + | 0.0565656i | 0 | −0.0144179 | + | 0.0249726i | 0 | 1.51695 | − | 2.58822i | 0 | ||||||||||
89.3 | 0 | −1.35104 | − | 1.08383i | 0 | 3.33482 | − | 0.588019i | 0 | 2.19892 | − | 3.80864i | 0 | 0.650612 | + | 2.92860i | 0 | ||||||||||
89.4 | 0 | −0.0815079 | − | 1.73013i | 0 | −1.23805 | + | 0.218301i | 0 | −0.753669 | + | 1.30539i | 0 | −2.98671 | + | 0.282039i | 0 | ||||||||||
89.5 | 0 | 0.144474 | + | 1.72601i | 0 | 2.45783 | − | 0.433381i | 0 | −2.29577 | + | 3.97638i | 0 | −2.95825 | + | 0.498730i | 0 | ||||||||||
89.6 | 0 | 0.264929 | + | 1.71167i | 0 | −4.07516 | + | 0.718561i | 0 | 0.904376 | − | 1.56643i | 0 | −2.85963 | + | 0.906941i | 0 | ||||||||||
89.7 | 0 | 0.505407 | − | 1.65667i | 0 | 2.78114 | − | 0.490390i | 0 | −1.12097 | + | 1.94159i | 0 | −2.48913 | − | 1.67459i | 0 | ||||||||||
89.8 | 0 | 1.20462 | + | 1.24454i | 0 | 1.70062 | − | 0.299865i | 0 | 1.13747 | − | 1.97015i | 0 | −0.0977739 | + | 2.99841i | 0 | ||||||||||
89.9 | 0 | 1.57581 | − | 0.718900i | 0 | 0.726176 | − | 0.128044i | 0 | 1.38478 | − | 2.39851i | 0 | 1.96636 | − | 2.26570i | 0 | ||||||||||
89.10 | 0 | 1.73038 | + | 0.0760235i | 0 | −2.36417 | + | 0.416867i | 0 | −2.12924 | + | 3.68795i | 0 | 2.98844 | + | 0.263099i | 0 | ||||||||||
See all 60 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
57.j | even | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 456.2.bm.b | yes | 60 |
3.b | odd | 2 | 1 | 456.2.bm.a | ✓ | 60 | |
4.b | odd | 2 | 1 | 912.2.cc.g | 60 | ||
12.b | even | 2 | 1 | 912.2.cc.h | 60 | ||
19.f | odd | 18 | 1 | 456.2.bm.a | ✓ | 60 | |
57.j | even | 18 | 1 | inner | 456.2.bm.b | yes | 60 |
76.k | even | 18 | 1 | 912.2.cc.h | 60 | ||
228.u | odd | 18 | 1 | 912.2.cc.g | 60 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
456.2.bm.a | ✓ | 60 | 3.b | odd | 2 | 1 | |
456.2.bm.a | ✓ | 60 | 19.f | odd | 18 | 1 | |
456.2.bm.b | yes | 60 | 1.a | even | 1 | 1 | trivial |
456.2.bm.b | yes | 60 | 57.j | even | 18 | 1 | inner |
912.2.cc.g | 60 | 4.b | odd | 2 | 1 | ||
912.2.cc.g | 60 | 228.u | odd | 18 | 1 | ||
912.2.cc.h | 60 | 12.b | even | 2 | 1 | ||
912.2.cc.h | 60 | 76.k | even | 18 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{60} - 3 T_{5}^{58} + 36 T_{5}^{57} + 3 T_{5}^{56} - 78 T_{5}^{55} - 3824 T_{5}^{54} + \cdots + 66324791296 \) acting on \(S_{2}^{\mathrm{new}}(456, [\chi])\).