Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [504,2,Mod(193,504)]
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, 0, 2, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("504.193");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 504.t (of order \(3\), 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: | \(22\) |
Relative dimension: | \(11\) over \(\Q(\zeta_{3})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
193.1 | 0 | −1.72692 | + | 0.133195i | 0 | −4.22296 | 0 | −2.37802 | + | 1.15974i | 0 | 2.96452 | − | 0.460034i | 0 | ||||||||||||
193.2 | 0 | −1.72608 | − | 0.143720i | 0 | 2.77180 | 0 | 0.855737 | + | 2.50354i | 0 | 2.95869 | + | 0.496145i | 0 | ||||||||||||
193.3 | 0 | −1.07968 | + | 1.35436i | 0 | 3.40736 | 0 | −2.05842 | − | 1.66220i | 0 | −0.668594 | − | 2.92455i | 0 | ||||||||||||
193.4 | 0 | −1.04208 | − | 1.38350i | 0 | −0.0619693 | 0 | −1.63689 | − | 2.07860i | 0 | −0.828124 | + | 2.88344i | 0 | ||||||||||||
193.5 | 0 | −0.816621 | + | 1.52746i | 0 | −1.78355 | 0 | 1.90167 | − | 1.83948i | 0 | −1.66626 | − | 2.49471i | 0 | ||||||||||||
193.6 | 0 | −0.341725 | − | 1.69801i | 0 | 0.526004 | 0 | 2.43963 | + | 1.02383i | 0 | −2.76645 | + | 1.16050i | 0 | ||||||||||||
193.7 | 0 | 0.666060 | + | 1.59886i | 0 | −0.468169 | 0 | −2.39007 | − | 1.13471i | 0 | −2.11273 | + | 2.12988i | 0 | ||||||||||||
193.8 | 0 | 0.677409 | − | 1.59409i | 0 | −2.66851 | 0 | −0.654882 | + | 2.56342i | 0 | −2.08224 | − | 2.15970i | 0 | ||||||||||||
193.9 | 0 | 1.13766 | + | 1.30604i | 0 | 3.19500 | 0 | 2.61289 | − | 0.415693i | 0 | −0.411479 | + | 2.97165i | 0 | ||||||||||||
193.10 | 0 | 1.52946 | + | 0.812868i | 0 | −3.79940 | 0 | 2.59312 | + | 0.525101i | 0 | 1.67849 | + | 2.48650i | 0 | ||||||||||||
193.11 | 0 | 1.72252 | − | 0.181425i | 0 | 2.10440 | 0 | −1.78475 | + | 1.95312i | 0 | 2.93417 | − | 0.625017i | 0 | ||||||||||||
457.1 | 0 | −1.72692 | − | 0.133195i | 0 | −4.22296 | 0 | −2.37802 | − | 1.15974i | 0 | 2.96452 | + | 0.460034i | 0 | ||||||||||||
457.2 | 0 | −1.72608 | + | 0.143720i | 0 | 2.77180 | 0 | 0.855737 | − | 2.50354i | 0 | 2.95869 | − | 0.496145i | 0 | ||||||||||||
457.3 | 0 | −1.07968 | − | 1.35436i | 0 | 3.40736 | 0 | −2.05842 | + | 1.66220i | 0 | −0.668594 | + | 2.92455i | 0 | ||||||||||||
457.4 | 0 | −1.04208 | + | 1.38350i | 0 | −0.0619693 | 0 | −1.63689 | + | 2.07860i | 0 | −0.828124 | − | 2.88344i | 0 | ||||||||||||
457.5 | 0 | −0.816621 | − | 1.52746i | 0 | −1.78355 | 0 | 1.90167 | + | 1.83948i | 0 | −1.66626 | + | 2.49471i | 0 | ||||||||||||
457.6 | 0 | −0.341725 | + | 1.69801i | 0 | 0.526004 | 0 | 2.43963 | − | 1.02383i | 0 | −2.76645 | − | 1.16050i | 0 | ||||||||||||
457.7 | 0 | 0.666060 | − | 1.59886i | 0 | −0.468169 | 0 | −2.39007 | + | 1.13471i | 0 | −2.11273 | − | 2.12988i | 0 | ||||||||||||
457.8 | 0 | 0.677409 | + | 1.59409i | 0 | −2.66851 | 0 | −0.654882 | − | 2.56342i | 0 | −2.08224 | + | 2.15970i | 0 | ||||||||||||
457.9 | 0 | 1.13766 | − | 1.30604i | 0 | 3.19500 | 0 | 2.61289 | + | 0.415693i | 0 | −0.411479 | − | 2.97165i | 0 | ||||||||||||
See all 22 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
63.g | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 504.2.t.c | yes | 22 |
3.b | odd | 2 | 1 | 1512.2.t.c | 22 | ||
4.b | odd | 2 | 1 | 1008.2.t.l | 22 | ||
7.c | even | 3 | 1 | 504.2.q.c | ✓ | 22 | |
9.c | even | 3 | 1 | 504.2.q.c | ✓ | 22 | |
9.d | odd | 6 | 1 | 1512.2.q.d | 22 | ||
12.b | even | 2 | 1 | 3024.2.t.k | 22 | ||
21.h | odd | 6 | 1 | 1512.2.q.d | 22 | ||
28.g | odd | 6 | 1 | 1008.2.q.l | 22 | ||
36.f | odd | 6 | 1 | 1008.2.q.l | 22 | ||
36.h | even | 6 | 1 | 3024.2.q.l | 22 | ||
63.g | even | 3 | 1 | inner | 504.2.t.c | yes | 22 |
63.n | odd | 6 | 1 | 1512.2.t.c | 22 | ||
84.n | even | 6 | 1 | 3024.2.q.l | 22 | ||
252.o | even | 6 | 1 | 3024.2.t.k | 22 | ||
252.bl | odd | 6 | 1 | 1008.2.t.l | 22 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
504.2.q.c | ✓ | 22 | 7.c | even | 3 | 1 | |
504.2.q.c | ✓ | 22 | 9.c | even | 3 | 1 | |
504.2.t.c | yes | 22 | 1.a | even | 1 | 1 | trivial |
504.2.t.c | yes | 22 | 63.g | even | 3 | 1 | inner |
1008.2.q.l | 22 | 28.g | odd | 6 | 1 | ||
1008.2.q.l | 22 | 36.f | odd | 6 | 1 | ||
1008.2.t.l | 22 | 4.b | odd | 2 | 1 | ||
1008.2.t.l | 22 | 252.bl | odd | 6 | 1 | ||
1512.2.q.d | 22 | 9.d | odd | 6 | 1 | ||
1512.2.q.d | 22 | 21.h | odd | 6 | 1 | ||
1512.2.t.c | 22 | 3.b | odd | 2 | 1 | ||
1512.2.t.c | 22 | 63.n | odd | 6 | 1 | ||
3024.2.q.l | 22 | 36.h | even | 6 | 1 | ||
3024.2.q.l | 22 | 84.n | even | 6 | 1 | ||
3024.2.t.k | 22 | 12.b | even | 2 | 1 | ||
3024.2.t.k | 22 | 252.o | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{11} + T_{5}^{10} - 38 T_{5}^{9} - 21 T_{5}^{8} + 513 T_{5}^{7} + 108 T_{5}^{6} - 2841 T_{5}^{5} + \cdots - 74 \) acting on \(S_{2}^{\mathrm{new}}(504, [\chi])\).