Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [576,3,Mod(353,576)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(576, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 3, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("576.353");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 576 = 2^{6} \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 576.n (of order \(6\), degree \(2\), 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: | \(15.6948632272\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
353.1 | 0 | −2.90962 | − | 0.730846i | 0 | −3.09427 | − | 5.35943i | 0 | −0.183090 | + | 0.317122i | 0 | 7.93173 | + | 4.25296i | 0 | ||||||||||
353.2 | 0 | −2.81879 | + | 1.02686i | 0 | 1.79985 | + | 3.11744i | 0 | −3.70215 | + | 6.41232i | 0 | 6.89112 | − | 5.78899i | 0 | ||||||||||
353.3 | 0 | −2.77167 | − | 1.14797i | 0 | −1.13094 | − | 1.95884i | 0 | −2.43128 | + | 4.21109i | 0 | 6.36431 | + | 6.36361i | 0 | ||||||||||
353.4 | 0 | −2.72431 | + | 1.25624i | 0 | 4.89158 | + | 8.47246i | 0 | 3.88386 | − | 6.72705i | 0 | 5.84373 | − | 6.84477i | 0 | ||||||||||
353.5 | 0 | −1.32664 | − | 2.69073i | 0 | 2.27609 | + | 3.94230i | 0 | 4.81066 | − | 8.33232i | 0 | −5.48007 | + | 7.13924i | 0 | ||||||||||
353.6 | 0 | −1.21852 | + | 2.74139i | 0 | −2.39677 | − | 4.15133i | 0 | 1.12793 | − | 1.95364i | 0 | −6.03041 | − | 6.68088i | 0 | ||||||||||
353.7 | 0 | −1.08485 | + | 2.79698i | 0 | 3.07518 | + | 5.32637i | 0 | 5.35886 | − | 9.28182i | 0 | −6.64619 | − | 6.06862i | 0 | ||||||||||
353.8 | 0 | −0.559363 | − | 2.94739i | 0 | −0.920717 | − | 1.59473i | 0 | 4.56979 | − | 7.91511i | 0 | −8.37423 | + | 3.29733i | 0 | ||||||||||
353.9 | 0 | 0.559363 | + | 2.94739i | 0 | −0.920717 | − | 1.59473i | 0 | −4.56979 | + | 7.91511i | 0 | −8.37423 | + | 3.29733i | 0 | ||||||||||
353.10 | 0 | 1.08485 | − | 2.79698i | 0 | 3.07518 | + | 5.32637i | 0 | −5.35886 | + | 9.28182i | 0 | −6.64619 | − | 6.06862i | 0 | ||||||||||
353.11 | 0 | 1.21852 | − | 2.74139i | 0 | −2.39677 | − | 4.15133i | 0 | −1.12793 | + | 1.95364i | 0 | −6.03041 | − | 6.68088i | 0 | ||||||||||
353.12 | 0 | 1.32664 | + | 2.69073i | 0 | 2.27609 | + | 3.94230i | 0 | −4.81066 | + | 8.33232i | 0 | −5.48007 | + | 7.13924i | 0 | ||||||||||
353.13 | 0 | 2.72431 | − | 1.25624i | 0 | 4.89158 | + | 8.47246i | 0 | −3.88386 | + | 6.72705i | 0 | 5.84373 | − | 6.84477i | 0 | ||||||||||
353.14 | 0 | 2.77167 | + | 1.14797i | 0 | −1.13094 | − | 1.95884i | 0 | 2.43128 | − | 4.21109i | 0 | 6.36431 | + | 6.36361i | 0 | ||||||||||
353.15 | 0 | 2.81879 | − | 1.02686i | 0 | 1.79985 | + | 3.11744i | 0 | 3.70215 | − | 6.41232i | 0 | 6.89112 | − | 5.78899i | 0 | ||||||||||
353.16 | 0 | 2.90962 | + | 0.730846i | 0 | −3.09427 | − | 5.35943i | 0 | 0.183090 | − | 0.317122i | 0 | 7.93173 | + | 4.25296i | 0 | ||||||||||
545.1 | 0 | −2.90962 | + | 0.730846i | 0 | −3.09427 | + | 5.35943i | 0 | −0.183090 | − | 0.317122i | 0 | 7.93173 | − | 4.25296i | 0 | ||||||||||
545.2 | 0 | −2.81879 | − | 1.02686i | 0 | 1.79985 | − | 3.11744i | 0 | −3.70215 | − | 6.41232i | 0 | 6.89112 | + | 5.78899i | 0 | ||||||||||
545.3 | 0 | −2.77167 | + | 1.14797i | 0 | −1.13094 | + | 1.95884i | 0 | −2.43128 | − | 4.21109i | 0 | 6.36431 | − | 6.36361i | 0 | ||||||||||
545.4 | 0 | −2.72431 | − | 1.25624i | 0 | 4.89158 | − | 8.47246i | 0 | 3.88386 | + | 6.72705i | 0 | 5.84373 | + | 6.84477i | 0 | ||||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
72.j | odd | 6 | 1 | inner |
72.l | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 576.3.n.d | yes | 32 |
3.b | odd | 2 | 1 | 1728.3.n.c | 32 | ||
4.b | odd | 2 | 1 | inner | 576.3.n.d | yes | 32 |
8.b | even | 2 | 1 | 576.3.n.c | ✓ | 32 | |
8.d | odd | 2 | 1 | 576.3.n.c | ✓ | 32 | |
9.c | even | 3 | 1 | 1728.3.n.d | 32 | ||
9.d | odd | 6 | 1 | 576.3.n.c | ✓ | 32 | |
12.b | even | 2 | 1 | 1728.3.n.c | 32 | ||
24.f | even | 2 | 1 | 1728.3.n.d | 32 | ||
24.h | odd | 2 | 1 | 1728.3.n.d | 32 | ||
36.f | odd | 6 | 1 | 1728.3.n.d | 32 | ||
36.h | even | 6 | 1 | 576.3.n.c | ✓ | 32 | |
72.j | odd | 6 | 1 | inner | 576.3.n.d | yes | 32 |
72.l | even | 6 | 1 | inner | 576.3.n.d | yes | 32 |
72.n | even | 6 | 1 | 1728.3.n.c | 32 | ||
72.p | odd | 6 | 1 | 1728.3.n.c | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
576.3.n.c | ✓ | 32 | 8.b | even | 2 | 1 | |
576.3.n.c | ✓ | 32 | 8.d | odd | 2 | 1 | |
576.3.n.c | ✓ | 32 | 9.d | odd | 6 | 1 | |
576.3.n.c | ✓ | 32 | 36.h | even | 6 | 1 | |
576.3.n.d | yes | 32 | 1.a | even | 1 | 1 | trivial |
576.3.n.d | yes | 32 | 4.b | odd | 2 | 1 | inner |
576.3.n.d | yes | 32 | 72.j | odd | 6 | 1 | inner |
576.3.n.d | yes | 32 | 72.l | even | 6 | 1 | inner |
1728.3.n.c | 32 | 3.b | odd | 2 | 1 | ||
1728.3.n.c | 32 | 12.b | even | 2 | 1 | ||
1728.3.n.c | 32 | 72.n | even | 6 | 1 | ||
1728.3.n.c | 32 | 72.p | odd | 6 | 1 | ||
1728.3.n.d | 32 | 9.c | even | 3 | 1 | ||
1728.3.n.d | 32 | 24.f | even | 2 | 1 | ||
1728.3.n.d | 32 | 24.h | odd | 2 | 1 | ||
1728.3.n.d | 32 | 36.f | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} - 9 T_{5}^{15} + 159 T_{5}^{14} - 558 T_{5}^{13} + 9729 T_{5}^{12} - 25515 T_{5}^{11} + \cdots + 14841086976 \) acting on \(S_{3}^{\mathrm{new}}(576, [\chi])\).