Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [441,2,Mod(17,441)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(441, base_ring=CyclotomicField(42))
chi = DirichletCharacter(H, H._module([21, 25]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("441.17");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 441 = 3^{2} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 441.bg (of order \(42\), degree \(12\), 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: | \(3.52140272914\) |
Analytic rank: | \(0\) |
Dimension: | \(216\) |
Relative dimension: | \(18\) over \(\Q(\zeta_{42})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{42}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 | −2.68471 | + | 0.201191i | 0 | 5.18952 | − | 0.782194i | −0.00530717 | + | 0.00492433i | 0 | −1.62374 | − | 2.08889i | −8.52551 | + | 1.94589i | 0 | 0.0132575 | − | 0.0142882i | ||||||
17.2 | −2.47614 | + | 0.185561i | 0 | 4.11916 | − | 0.620863i | 1.08753 | − | 1.00908i | 0 | 0.513890 | − | 2.59536i | −5.24274 | + | 1.19662i | 0 | −2.50562 | + | 2.70041i | ||||||
17.3 | −2.05710 | + | 0.154158i | 0 | 2.23024 | − | 0.336155i | −1.16105 | + | 1.07730i | 0 | 1.82087 | + | 1.91949i | −0.513717 | + | 0.117252i | 0 | 2.22233 | − | 2.39510i | ||||||
17.4 | −1.57786 | + | 0.118244i | 0 | 0.497985 | − | 0.0750592i | −2.70933 | + | 2.51389i | 0 | 1.38562 | − | 2.25390i | 2.30834 | − | 0.526865i | 0 | 3.97769 | − | 4.28693i | ||||||
17.5 | −1.51461 | + | 0.113504i | 0 | 0.303503 | − | 0.0457458i | 2.45092 | − | 2.27412i | 0 | 1.78982 | + | 1.94847i | 2.50706 | − | 0.572220i | 0 | −3.45407 | + | 3.72260i | ||||||
17.6 | −1.49398 | + | 0.111958i | 0 | 0.241771 | − | 0.0364412i | 1.96165 | − | 1.82014i | 0 | −2.52569 | + | 0.787950i | 2.56409 | − | 0.585236i | 0 | −2.72688 | + | 2.93888i | ||||||
17.7 | −0.859142 | + | 0.0643838i | 0 | −1.24368 | + | 0.187455i | 0.593873 | − | 0.551034i | 0 | −2.64451 | − | 0.0809471i | 2.73633 | − | 0.624550i | 0 | −0.474744 | + | 0.511652i | ||||||
17.8 | −0.601956 | + | 0.0451104i | 0 | −1.61735 | + | 0.243776i | −1.12371 | + | 1.04265i | 0 | −0.186554 | − | 2.63917i | 2.13959 | − | 0.488348i | 0 | 0.629389 | − | 0.678320i | ||||||
17.9 | −0.409998 | + | 0.0307251i | 0 | −1.81051 | + | 0.272890i | −2.06043 | + | 1.91180i | 0 | −0.0705282 | + | 2.64481i | 1.53560 | − | 0.350490i | 0 | 0.786031 | − | 0.847140i | ||||||
17.10 | 0.409998 | − | 0.0307251i | 0 | −1.81051 | + | 0.272890i | 2.06043 | − | 1.91180i | 0 | −0.0705282 | + | 2.64481i | −1.53560 | + | 0.350490i | 0 | 0.786031 | − | 0.847140i | ||||||
17.11 | 0.601956 | − | 0.0451104i | 0 | −1.61735 | + | 0.243776i | 1.12371 | − | 1.04265i | 0 | −0.186554 | − | 2.63917i | −2.13959 | + | 0.488348i | 0 | 0.629389 | − | 0.678320i | ||||||
17.12 | 0.859142 | − | 0.0643838i | 0 | −1.24368 | + | 0.187455i | −0.593873 | + | 0.551034i | 0 | −2.64451 | − | 0.0809471i | −2.73633 | + | 0.624550i | 0 | −0.474744 | + | 0.511652i | ||||||
17.13 | 1.49398 | − | 0.111958i | 0 | 0.241771 | − | 0.0364412i | −1.96165 | + | 1.82014i | 0 | −2.52569 | + | 0.787950i | −2.56409 | + | 0.585236i | 0 | −2.72688 | + | 2.93888i | ||||||
17.14 | 1.51461 | − | 0.113504i | 0 | 0.303503 | − | 0.0457458i | −2.45092 | + | 2.27412i | 0 | 1.78982 | + | 1.94847i | −2.50706 | + | 0.572220i | 0 | −3.45407 | + | 3.72260i | ||||||
17.15 | 1.57786 | − | 0.118244i | 0 | 0.497985 | − | 0.0750592i | 2.70933 | − | 2.51389i | 0 | 1.38562 | − | 2.25390i | −2.30834 | + | 0.526865i | 0 | 3.97769 | − | 4.28693i | ||||||
17.16 | 2.05710 | − | 0.154158i | 0 | 2.23024 | − | 0.336155i | 1.16105 | − | 1.07730i | 0 | 1.82087 | + | 1.91949i | 0.513717 | − | 0.117252i | 0 | 2.22233 | − | 2.39510i | ||||||
17.17 | 2.47614 | − | 0.185561i | 0 | 4.11916 | − | 0.620863i | −1.08753 | + | 1.00908i | 0 | 0.513890 | − | 2.59536i | 5.24274 | − | 1.19662i | 0 | −2.50562 | + | 2.70041i | ||||||
17.18 | 2.68471 | − | 0.201191i | 0 | 5.18952 | − | 0.782194i | 0.00530717 | − | 0.00492433i | 0 | −1.62374 | − | 2.08889i | 8.52551 | − | 1.94589i | 0 | 0.0132575 | − | 0.0142882i | ||||||
26.1 | −2.68471 | − | 0.201191i | 0 | 5.18952 | + | 0.782194i | −0.00530717 | − | 0.00492433i | 0 | −1.62374 | + | 2.08889i | −8.52551 | − | 1.94589i | 0 | 0.0132575 | + | 0.0142882i | ||||||
26.2 | −2.47614 | − | 0.185561i | 0 | 4.11916 | + | 0.620863i | 1.08753 | + | 1.00908i | 0 | 0.513890 | + | 2.59536i | −5.24274 | − | 1.19662i | 0 | −2.50562 | − | 2.70041i | ||||||
See next 80 embeddings (of 216 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
49.h | odd | 42 | 1 | inner |
147.o | even | 42 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 441.2.bg.a | ✓ | 216 |
3.b | odd | 2 | 1 | inner | 441.2.bg.a | ✓ | 216 |
49.h | odd | 42 | 1 | inner | 441.2.bg.a | ✓ | 216 |
147.o | even | 42 | 1 | inner | 441.2.bg.a | ✓ | 216 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
441.2.bg.a | ✓ | 216 | 1.a | even | 1 | 1 | trivial |
441.2.bg.a | ✓ | 216 | 3.b | odd | 2 | 1 | inner |
441.2.bg.a | ✓ | 216 | 49.h | odd | 42 | 1 | inner |
441.2.bg.a | ✓ | 216 | 147.o | even | 42 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(441, [\chi])\).