Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [441,3,Mod(31,441)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(441, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("441.31");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 441 = 3^{2} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 441.k (of order \(6\), 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: | \(12.0163796583\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Relative dimension: | \(14\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 63) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
31.1 | −1.67789 | + | 2.90618i | −2.39606 | + | 1.80524i | −3.63061 | − | 6.28839i | − | 8.51666i | −1.22605 | − | 9.99238i | 0 | 10.9439 | 2.48221 | − | 8.65093i | 24.7510 | + | 14.2900i | |||||
31.2 | −1.67756 | + | 2.90562i | 2.99109 | + | 0.230993i | −3.62842 | − | 6.28461i | − | 0.888628i | −5.68892 | + | 8.30348i | 0 | 10.9271 | 8.89328 | + | 1.38184i | 2.58202 | + | 1.49073i | |||||
31.3 | −1.32841 | + | 2.30087i | −0.187748 | + | 2.99412i | −1.52933 | − | 2.64888i | 9.20400i | −6.63967 | − | 4.40939i | 0 | −2.50096 | −8.92950 | − | 1.12428i | −21.1772 | − | 12.2267i | ||||||
31.4 | −1.12025 | + | 1.94033i | −1.92745 | − | 2.29890i | −0.509909 | − | 0.883189i | − | 1.93444i | 6.61983 | − | 1.16455i | 0 | −6.67708 | −1.56985 | + | 8.86203i | 3.75345 | + | 2.16706i | |||||
31.5 | −0.840995 | + | 1.45665i | 1.62217 | − | 2.52360i | 0.585454 | + | 1.01404i | 2.34462i | 2.31176 | + | 4.48526i | 0 | −8.69742 | −3.73715 | − | 8.18741i | −3.41529 | − | 1.97182i | ||||||
31.6 | −0.227576 | + | 0.394173i | 2.69941 | + | 1.30889i | 1.89642 | + | 3.28469i | − | 4.37081i | −1.13025 | + | 0.766164i | 0 | −3.54692 | 5.57364 | + | 7.06644i | 1.72285 | + | 0.994690i | |||||
31.7 | −0.178911 | + | 0.309883i | −2.99942 | + | 0.0589959i | 1.93598 | + | 3.35322i | 4.59004i | 0.518348 | − | 0.940026i | 0 | −2.81677 | 8.99304 | − | 0.353907i | −1.42238 | − | 0.821210i | ||||||
31.8 | 0.198068 | − | 0.343064i | −1.78343 | + | 2.41234i | 1.92154 | + | 3.32820i | − | 2.97240i | 0.474346 | + | 1.08964i | 0 | 3.10693 | −2.63876 | − | 8.60447i | −1.01972 | − | 0.588737i | |||||
31.9 | 0.662399 | − | 1.14731i | 1.57692 | − | 2.55212i | 1.12246 | + | 1.94415i | − | 7.23514i | −1.88353 | − | 3.49973i | 0 | 8.27324 | −4.02667 | − | 8.04897i | −8.30093 | − | 4.79254i | |||||
31.10 | 0.826674 | − | 1.43184i | 2.53185 | + | 1.60927i | 0.633221 | + | 1.09677i | 7.86923i | 4.39723 | − | 2.29486i | 0 | 8.70726 | 3.82051 | + | 8.14885i | 11.2675 | + | 6.50529i | ||||||
31.11 | 0.902282 | − | 1.56280i | −0.540538 | − | 2.95090i | 0.371774 | + | 0.643931i | 5.75495i | −5.09938 | − | 1.81779i | 0 | 8.56004 | −8.41564 | + | 3.19015i | 8.99383 | + | 5.19259i | ||||||
31.12 | 1.41697 | − | 2.45427i | −0.222472 | + | 2.99174i | −2.01561 | − | 3.49114i | − | 2.39855i | 7.02729 | + | 4.78521i | 0 | −0.0884848 | −8.90101 | − | 1.33116i | −5.88667 | − | 3.39867i | |||||
31.13 | 1.62718 | − | 2.81835i | −2.71323 | − | 1.27999i | −3.29541 | − | 5.70782i | − | 3.39483i | −8.02237 | + | 5.56407i | 0 | −8.43145 | 5.72324 | + | 6.94582i | −9.56782 | − | 5.52398i | |||||
31.14 | 1.91801 | − | 3.32210i | 2.84892 | − | 0.940039i | −5.35756 | − | 9.27956i | 0.216546i | 2.34136 | − | 11.2674i | 0 | −25.7594 | 7.23265 | − | 5.35619i | 0.719386 | + | 0.415338i | ||||||
313.1 | −1.67789 | − | 2.90618i | −2.39606 | − | 1.80524i | −3.63061 | + | 6.28839i | 8.51666i | −1.22605 | + | 9.99238i | 0 | 10.9439 | 2.48221 | + | 8.65093i | 24.7510 | − | 14.2900i | ||||||
313.2 | −1.67756 | − | 2.90562i | 2.99109 | − | 0.230993i | −3.62842 | + | 6.28461i | 0.888628i | −5.68892 | − | 8.30348i | 0 | 10.9271 | 8.89328 | − | 1.38184i | 2.58202 | − | 1.49073i | ||||||
313.3 | −1.32841 | − | 2.30087i | −0.187748 | − | 2.99412i | −1.52933 | + | 2.64888i | − | 9.20400i | −6.63967 | + | 4.40939i | 0 | −2.50096 | −8.92950 | + | 1.12428i | −21.1772 | + | 12.2267i | |||||
313.4 | −1.12025 | − | 1.94033i | −1.92745 | + | 2.29890i | −0.509909 | + | 0.883189i | 1.93444i | 6.61983 | + | 1.16455i | 0 | −6.67708 | −1.56985 | − | 8.86203i | 3.75345 | − | 2.16706i | ||||||
313.5 | −0.840995 | − | 1.45665i | 1.62217 | + | 2.52360i | 0.585454 | − | 1.01404i | − | 2.34462i | 2.31176 | − | 4.48526i | 0 | −8.69742 | −3.73715 | + | 8.18741i | −3.41529 | + | 1.97182i | |||||
313.6 | −0.227576 | − | 0.394173i | 2.69941 | − | 1.30889i | 1.89642 | − | 3.28469i | 4.37081i | −1.13025 | − | 0.766164i | 0 | −3.54692 | 5.57364 | − | 7.06644i | 1.72285 | − | 0.994690i | ||||||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
63.k | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 441.3.k.b | 28 | |
7.b | odd | 2 | 1 | 63.3.k.a | ✓ | 28 | |
7.c | even | 3 | 1 | 63.3.t.a | yes | 28 | |
7.c | even | 3 | 1 | 441.3.l.b | 28 | ||
7.d | odd | 6 | 1 | 441.3.l.a | 28 | ||
7.d | odd | 6 | 1 | 441.3.t.a | 28 | ||
9.c | even | 3 | 1 | 441.3.t.a | 28 | ||
21.c | even | 2 | 1 | 189.3.k.a | 28 | ||
21.h | odd | 6 | 1 | 189.3.t.a | 28 | ||
63.g | even | 3 | 1 | 63.3.k.a | ✓ | 28 | |
63.h | even | 3 | 1 | 441.3.l.a | 28 | ||
63.k | odd | 6 | 1 | inner | 441.3.k.b | 28 | |
63.l | odd | 6 | 1 | 63.3.t.a | yes | 28 | |
63.n | odd | 6 | 1 | 189.3.k.a | 28 | ||
63.o | even | 6 | 1 | 189.3.t.a | 28 | ||
63.t | odd | 6 | 1 | 441.3.l.b | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
63.3.k.a | ✓ | 28 | 7.b | odd | 2 | 1 | |
63.3.k.a | ✓ | 28 | 63.g | even | 3 | 1 | |
63.3.t.a | yes | 28 | 7.c | even | 3 | 1 | |
63.3.t.a | yes | 28 | 63.l | odd | 6 | 1 | |
189.3.k.a | 28 | 21.c | even | 2 | 1 | ||
189.3.k.a | 28 | 63.n | odd | 6 | 1 | ||
189.3.t.a | 28 | 21.h | odd | 6 | 1 | ||
189.3.t.a | 28 | 63.o | even | 6 | 1 | ||
441.3.k.b | 28 | 1.a | even | 1 | 1 | trivial | |
441.3.k.b | 28 | 63.k | odd | 6 | 1 | inner | |
441.3.l.a | 28 | 7.d | odd | 6 | 1 | ||
441.3.l.a | 28 | 63.h | even | 3 | 1 | ||
441.3.l.b | 28 | 7.c | even | 3 | 1 | ||
441.3.l.b | 28 | 63.t | odd | 6 | 1 | ||
441.3.t.a | 28 | 7.d | odd | 6 | 1 | ||
441.3.t.a | 28 | 9.c | even | 3 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{28} - T_{2}^{27} + 40 T_{2}^{26} - 29 T_{2}^{25} + 977 T_{2}^{24} - 620 T_{2}^{23} + \cdots + 1034289 \)
acting on \(S_{3}^{\mathrm{new}}(441, [\chi])\).