Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [504,2,Mod(269,504)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(504, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 3, 3, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("504.269");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 504.ch (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: | \(4.02446026187\) |
Analytic rank: | \(0\) |
Dimension: | \(56\) |
Relative dimension: | \(28\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
269.1 | −1.41410 | + | 0.0182389i | 0 | 1.99933 | − | 0.0515832i | −0.785247 | + | 0.453362i | 0 | −2.47043 | − | 0.947077i | −2.82631 | + | 0.109409i | 0 | 1.10215 | − | 0.655420i | ||||||
269.2 | −1.40843 | − | 0.127777i | 0 | 1.96735 | + | 0.359931i | 1.87230 | − | 1.08097i | 0 | 2.55958 | + | 0.669737i | −2.72488 | − | 0.758320i | 0 | −2.77512 | + | 1.28324i | ||||||
269.3 | −1.39540 | + | 0.229930i | 0 | 1.89426 | − | 0.641688i | −3.16007 | + | 1.82447i | 0 | −1.64838 | + | 2.06951i | −2.49571 | + | 1.33096i | 0 | 3.99006 | − | 3.27246i | ||||||
269.4 | −1.30255 | − | 0.550794i | 0 | 1.39325 | + | 1.43487i | −1.00441 | + | 0.579896i | 0 | 1.24394 | + | 2.33508i | −1.02446 | − | 2.63638i | 0 | 1.62769 | − | 0.202118i | ||||||
269.5 | −1.13135 | − | 0.848557i | 0 | 0.559903 | + | 1.92003i | −1.53798 | + | 0.887954i | 0 | 0.843933 | − | 2.50754i | 0.995807 | − | 2.64733i | 0 | 2.49347 | + | 0.300478i | ||||||
269.6 | −1.11071 | − | 0.875401i | 0 | 0.467346 | + | 1.94463i | 3.18706 | − | 1.84005i | 0 | −0.998380 | − | 2.45015i | 1.18325 | − | 2.56903i | 0 | −5.15068 | − | 0.746198i | ||||||
269.7 | −0.937852 | − | 1.05851i | 0 | −0.240867 | + | 1.98544i | −0.331990 | + | 0.191675i | 0 | −2.03027 | + | 1.69647i | 2.32750 | − | 1.60709i | 0 | 0.514246 | + | 0.171651i | ||||||
269.8 | −0.896824 | + | 1.09348i | 0 | −0.391414 | − | 1.96132i | 3.16007 | − | 1.82447i | 0 | −1.64838 | + | 2.06951i | 2.49571 | + | 1.33096i | 0 | −0.839002 | + | 5.09172i | ||||||
269.9 | −0.722843 | + | 1.21552i | 0 | −0.954995 | − | 1.75727i | 0.785247 | − | 0.453362i | 0 | −2.47043 | − | 0.947077i | 2.82631 | + | 0.109409i | 0 | −0.0165377 | + | 1.28220i | ||||||
269.10 | −0.593556 | + | 1.28362i | 0 | −1.29538 | − | 1.52381i | −1.87230 | + | 1.08097i | 0 | 2.55958 | + | 0.669737i | 2.72488 | − | 0.758320i | 0 | −0.276248 | − | 3.04495i | ||||||
269.11 | −0.447766 | − | 1.34146i | 0 | −1.59901 | + | 1.20132i | −0.331990 | + | 0.191675i | 0 | −2.03027 | + | 1.69647i | 2.32750 | + | 1.60709i | 0 | 0.405777 | + | 0.359525i | ||||||
269.12 | −0.202765 | − | 1.39960i | 0 | −1.91777 | + | 0.567582i | 3.18706 | − | 1.84005i | 0 | −0.998380 | − | 2.45015i | 1.18325 | + | 2.56903i | 0 | −3.22157 | − | 4.08752i | ||||||
269.13 | −0.174271 | + | 1.40343i | 0 | −1.93926 | − | 0.489157i | 1.00441 | − | 0.579896i | 0 | 1.24394 | + | 2.33508i | 1.02446 | − | 2.63638i | 0 | 0.638806 | + | 1.51068i | ||||||
269.14 | −0.169197 | − | 1.40406i | 0 | −1.94274 | + | 0.475124i | −1.53798 | + | 0.887954i | 0 | 0.843933 | − | 2.50754i | 0.995807 | + | 2.64733i | 0 | 1.50696 | + | 2.00917i | ||||||
269.15 | 0.169197 | + | 1.40406i | 0 | −1.94274 | + | 0.475124i | 1.53798 | − | 0.887954i | 0 | 0.843933 | − | 2.50754i | −0.995807 | − | 2.64733i | 0 | 1.50696 | + | 2.00917i | ||||||
269.16 | 0.174271 | − | 1.40343i | 0 | −1.93926 | − | 0.489157i | −1.00441 | + | 0.579896i | 0 | 1.24394 | + | 2.33508i | −1.02446 | + | 2.63638i | 0 | 0.638806 | + | 1.51068i | ||||||
269.17 | 0.202765 | + | 1.39960i | 0 | −1.91777 | + | 0.567582i | −3.18706 | + | 1.84005i | 0 | −0.998380 | − | 2.45015i | −1.18325 | − | 2.56903i | 0 | −3.22157 | − | 4.08752i | ||||||
269.18 | 0.447766 | + | 1.34146i | 0 | −1.59901 | + | 1.20132i | 0.331990 | − | 0.191675i | 0 | −2.03027 | + | 1.69647i | −2.32750 | − | 1.60709i | 0 | 0.405777 | + | 0.359525i | ||||||
269.19 | 0.593556 | − | 1.28362i | 0 | −1.29538 | − | 1.52381i | 1.87230 | − | 1.08097i | 0 | 2.55958 | + | 0.669737i | −2.72488 | + | 0.758320i | 0 | −0.276248 | − | 3.04495i | ||||||
269.20 | 0.722843 | − | 1.21552i | 0 | −0.954995 | − | 1.75727i | −0.785247 | + | 0.453362i | 0 | −2.47043 | − | 0.947077i | −2.82631 | − | 0.109409i | 0 | −0.0165377 | + | 1.28220i | ||||||
See all 56 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.d | odd | 6 | 1 | inner |
8.b | even | 2 | 1 | inner |
21.g | even | 6 | 1 | inner |
24.h | odd | 2 | 1 | inner |
56.j | odd | 6 | 1 | inner |
168.ba | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 504.2.ch.b | ✓ | 56 |
3.b | odd | 2 | 1 | inner | 504.2.ch.b | ✓ | 56 |
4.b | odd | 2 | 1 | 2016.2.cp.b | 56 | ||
7.d | odd | 6 | 1 | inner | 504.2.ch.b | ✓ | 56 |
8.b | even | 2 | 1 | inner | 504.2.ch.b | ✓ | 56 |
8.d | odd | 2 | 1 | 2016.2.cp.b | 56 | ||
12.b | even | 2 | 1 | 2016.2.cp.b | 56 | ||
21.g | even | 6 | 1 | inner | 504.2.ch.b | ✓ | 56 |
24.f | even | 2 | 1 | 2016.2.cp.b | 56 | ||
24.h | odd | 2 | 1 | inner | 504.2.ch.b | ✓ | 56 |
28.f | even | 6 | 1 | 2016.2.cp.b | 56 | ||
56.j | odd | 6 | 1 | inner | 504.2.ch.b | ✓ | 56 |
56.m | even | 6 | 1 | 2016.2.cp.b | 56 | ||
84.j | odd | 6 | 1 | 2016.2.cp.b | 56 | ||
168.ba | even | 6 | 1 | inner | 504.2.ch.b | ✓ | 56 |
168.be | odd | 6 | 1 | 2016.2.cp.b | 56 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
504.2.ch.b | ✓ | 56 | 1.a | even | 1 | 1 | trivial |
504.2.ch.b | ✓ | 56 | 3.b | odd | 2 | 1 | inner |
504.2.ch.b | ✓ | 56 | 7.d | odd | 6 | 1 | inner |
504.2.ch.b | ✓ | 56 | 8.b | even | 2 | 1 | inner |
504.2.ch.b | ✓ | 56 | 21.g | even | 6 | 1 | inner |
504.2.ch.b | ✓ | 56 | 24.h | odd | 2 | 1 | inner |
504.2.ch.b | ✓ | 56 | 56.j | odd | 6 | 1 | inner |
504.2.ch.b | ✓ | 56 | 168.ba | even | 6 | 1 | inner |
2016.2.cp.b | 56 | 4.b | odd | 2 | 1 | ||
2016.2.cp.b | 56 | 8.d | odd | 2 | 1 | ||
2016.2.cp.b | 56 | 12.b | even | 2 | 1 | ||
2016.2.cp.b | 56 | 24.f | even | 2 | 1 | ||
2016.2.cp.b | 56 | 28.f | even | 6 | 1 | ||
2016.2.cp.b | 56 | 56.m | even | 6 | 1 | ||
2016.2.cp.b | 56 | 84.j | odd | 6 | 1 | ||
2016.2.cp.b | 56 | 168.be | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{28} - 37 T_{5}^{26} + 882 T_{5}^{24} - 12429 T_{5}^{22} + 126330 T_{5}^{20} - 820581 T_{5}^{18} + \cdots + 186624 \) acting on \(S_{2}^{\mathrm{new}}(504, [\chi])\).