Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [576,4,Mod(145,576)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(576, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("576.145");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 576 = 2^{6} \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 576.k (of order \(4\), degree \(2\), not 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: | \(33.9851001633\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) over \(\Q(i)\) |
Twist minimal: | no (minimal twist has level 48) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
145.1 | 0 | 0 | 0 | −14.6111 | + | 14.6111i | 0 | − | 26.8889i | 0 | 0 | 0 | |||||||||||||||
145.2 | 0 | 0 | 0 | −11.7911 | + | 11.7911i | 0 | − | 12.5754i | 0 | 0 | 0 | |||||||||||||||
145.3 | 0 | 0 | 0 | −10.2951 | + | 10.2951i | 0 | 32.8369i | 0 | 0 | 0 | ||||||||||||||||
145.4 | 0 | 0 | 0 | −3.22588 | + | 3.22588i | 0 | 24.6080i | 0 | 0 | 0 | ||||||||||||||||
145.5 | 0 | 0 | 0 | −2.24191 | + | 2.24191i | 0 | 9.00196i | 0 | 0 | 0 | ||||||||||||||||
145.6 | 0 | 0 | 0 | −0.706564 | + | 0.706564i | 0 | − | 4.44122i | 0 | 0 | 0 | |||||||||||||||
145.7 | 0 | 0 | 0 | −0.644922 | + | 0.644922i | 0 | 7.13926i | 0 | 0 | 0 | ||||||||||||||||
145.8 | 0 | 0 | 0 | 3.72414 | − | 3.72414i | 0 | − | 20.2675i | 0 | 0 | 0 | |||||||||||||||
145.9 | 0 | 0 | 0 | 7.29121 | − | 7.29121i | 0 | − | 22.1610i | 0 | 0 | 0 | |||||||||||||||
145.10 | 0 | 0 | 0 | 8.83384 | − | 8.83384i | 0 | − | 29.4760i | 0 | 0 | 0 | |||||||||||||||
145.11 | 0 | 0 | 0 | 11.7719 | − | 11.7719i | 0 | 14.7089i | 0 | 0 | 0 | ||||||||||||||||
145.12 | 0 | 0 | 0 | 11.8955 | − | 11.8955i | 0 | − | 0.485059i | 0 | 0 | 0 | |||||||||||||||
433.1 | 0 | 0 | 0 | −14.6111 | − | 14.6111i | 0 | 26.8889i | 0 | 0 | 0 | ||||||||||||||||
433.2 | 0 | 0 | 0 | −11.7911 | − | 11.7911i | 0 | 12.5754i | 0 | 0 | 0 | ||||||||||||||||
433.3 | 0 | 0 | 0 | −10.2951 | − | 10.2951i | 0 | − | 32.8369i | 0 | 0 | 0 | |||||||||||||||
433.4 | 0 | 0 | 0 | −3.22588 | − | 3.22588i | 0 | − | 24.6080i | 0 | 0 | 0 | |||||||||||||||
433.5 | 0 | 0 | 0 | −2.24191 | − | 2.24191i | 0 | − | 9.00196i | 0 | 0 | 0 | |||||||||||||||
433.6 | 0 | 0 | 0 | −0.706564 | − | 0.706564i | 0 | 4.44122i | 0 | 0 | 0 | ||||||||||||||||
433.7 | 0 | 0 | 0 | −0.644922 | − | 0.644922i | 0 | − | 7.13926i | 0 | 0 | 0 | |||||||||||||||
433.8 | 0 | 0 | 0 | 3.72414 | + | 3.72414i | 0 | 20.2675i | 0 | 0 | 0 | ||||||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
16.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 576.4.k.b | 24 | |
3.b | odd | 2 | 1 | 192.4.j.a | 24 | ||
4.b | odd | 2 | 1 | 144.4.k.b | 24 | ||
12.b | even | 2 | 1 | 48.4.j.a | ✓ | 24 | |
16.e | even | 4 | 1 | inner | 576.4.k.b | 24 | |
16.f | odd | 4 | 1 | 144.4.k.b | 24 | ||
24.f | even | 2 | 1 | 384.4.j.b | 24 | ||
24.h | odd | 2 | 1 | 384.4.j.a | 24 | ||
48.i | odd | 4 | 1 | 192.4.j.a | 24 | ||
48.i | odd | 4 | 1 | 384.4.j.a | 24 | ||
48.k | even | 4 | 1 | 48.4.j.a | ✓ | 24 | |
48.k | even | 4 | 1 | 384.4.j.b | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
48.4.j.a | ✓ | 24 | 12.b | even | 2 | 1 | |
48.4.j.a | ✓ | 24 | 48.k | even | 4 | 1 | |
144.4.k.b | 24 | 4.b | odd | 2 | 1 | ||
144.4.k.b | 24 | 16.f | odd | 4 | 1 | ||
192.4.j.a | 24 | 3.b | odd | 2 | 1 | ||
192.4.j.a | 24 | 48.i | odd | 4 | 1 | ||
384.4.j.a | 24 | 24.h | odd | 2 | 1 | ||
384.4.j.a | 24 | 48.i | odd | 4 | 1 | ||
384.4.j.b | 24 | 24.f | even | 2 | 1 | ||
384.4.j.b | 24 | 48.k | even | 4 | 1 | ||
576.4.k.b | 24 | 1.a | even | 1 | 1 | trivial | |
576.4.k.b | 24 | 16.e | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - 1936 T_{5}^{21} + 249216 T_{5}^{20} - 752832 T_{5}^{19} + 1874048 T_{5}^{18} + \cdots + 15\!\cdots\!04 \) acting on \(S_{4}^{\mathrm{new}}(576, [\chi])\).