Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [261,3,Mod(53,261)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(261, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([7, 4]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("261.53");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 261 = 3^{2} \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 261.p (of order \(14\), degree \(6\), 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: | \(7.11173489980\) |
Analytic rank: | \(0\) |
Dimension: | \(120\) |
Relative dimension: | \(20\) over \(\Q(\zeta_{14})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{14}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
53.1 | −3.01719 | + | 2.40613i | 0 | 2.42390 | − | 10.6198i | −3.61293 | + | 2.88122i | 0 | 0.446262 | + | 1.95520i | 11.5416 | + | 23.9664i | 0 | 3.96833 | − | 17.3864i | ||||||
53.2 | −2.87082 | + | 2.28940i | 0 | 2.11016 | − | 9.24521i | −0.550556 | + | 0.439054i | 0 | 0.216354 | + | 0.947909i | 8.73539 | + | 18.1392i | 0 | 0.575376 | − | 2.52089i | ||||||
53.3 | −2.31293 | + | 1.84450i | 0 | 1.05738 | − | 4.63269i | 2.04793 | − | 1.63317i | 0 | −2.50930 | − | 10.9939i | 0.965025 | + | 2.00390i | 0 | −1.72433 | + | 7.55480i | ||||||
53.4 | −2.04594 | + | 1.63158i | 0 | 0.633728 | − | 2.77654i | 5.84586 | − | 4.66192i | 0 | 1.69806 | + | 7.43967i | −1.30805 | − | 2.71620i | 0 | −4.35398 | + | 19.0760i | ||||||
53.5 | −1.77322 | + | 1.41409i | 0 | 0.254554 | − | 1.11527i | −7.00899 | + | 5.58948i | 0 | −1.51727 | − | 6.64761i | −2.81052 | − | 5.83611i | 0 | 4.52440 | − | 19.8227i | ||||||
53.6 | −1.75160 | + | 1.39686i | 0 | 0.226820 | − | 0.993762i | 2.77979 | − | 2.21681i | 0 | 1.32518 | + | 5.80597i | −2.89742 | − | 6.01655i | 0 | −1.77253 | + | 7.76594i | ||||||
53.7 | −1.58966 | + | 1.26771i | 0 | 0.0298453 | − | 0.130761i | −5.19041 | + | 4.13921i | 0 | 1.56263 | + | 6.84632i | −3.41046 | − | 7.08190i | 0 | 3.00366 | − | 13.1599i | ||||||
53.8 | −0.865482 | + | 0.690199i | 0 | −0.617399 | + | 2.70500i | 1.36029 | − | 1.08479i | 0 | −1.33999 | − | 5.87088i | −3.25387 | − | 6.75673i | 0 | −0.428582 | + | 1.87774i | ||||||
53.9 | −0.504871 | + | 0.402621i | 0 | −0.797293 | + | 3.49317i | 4.97416 | − | 3.96676i | 0 | −2.49371 | − | 10.9257i | −2.12462 | − | 4.41182i | 0 | −0.914207 | + | 4.00540i | ||||||
53.10 | −0.227275 | + | 0.181246i | 0 | −0.871280 | + | 3.81733i | −1.73603 | + | 1.38444i | 0 | 1.50188 | + | 6.58016i | −0.998367 | − | 2.07313i | 0 | 0.143633 | − | 0.629297i | ||||||
53.11 | 0.227275 | − | 0.181246i | 0 | −0.871280 | + | 3.81733i | 1.73603 | − | 1.38444i | 0 | 1.50188 | + | 6.58016i | 0.998367 | + | 2.07313i | 0 | 0.143633 | − | 0.629297i | ||||||
53.12 | 0.504871 | − | 0.402621i | 0 | −0.797293 | + | 3.49317i | −4.97416 | + | 3.96676i | 0 | −2.49371 | − | 10.9257i | 2.12462 | + | 4.41182i | 0 | −0.914207 | + | 4.00540i | ||||||
53.13 | 0.865482 | − | 0.690199i | 0 | −0.617399 | + | 2.70500i | −1.36029 | + | 1.08479i | 0 | −1.33999 | − | 5.87088i | 3.25387 | + | 6.75673i | 0 | −0.428582 | + | 1.87774i | ||||||
53.14 | 1.58966 | − | 1.26771i | 0 | 0.0298453 | − | 0.130761i | 5.19041 | − | 4.13921i | 0 | 1.56263 | + | 6.84632i | 3.41046 | + | 7.08190i | 0 | 3.00366 | − | 13.1599i | ||||||
53.15 | 1.75160 | − | 1.39686i | 0 | 0.226820 | − | 0.993762i | −2.77979 | + | 2.21681i | 0 | 1.32518 | + | 5.80597i | 2.89742 | + | 6.01655i | 0 | −1.77253 | + | 7.76594i | ||||||
53.16 | 1.77322 | − | 1.41409i | 0 | 0.254554 | − | 1.11527i | 7.00899 | − | 5.58948i | 0 | −1.51727 | − | 6.64761i | 2.81052 | + | 5.83611i | 0 | 4.52440 | − | 19.8227i | ||||||
53.17 | 2.04594 | − | 1.63158i | 0 | 0.633728 | − | 2.77654i | −5.84586 | + | 4.66192i | 0 | 1.69806 | + | 7.43967i | 1.30805 | + | 2.71620i | 0 | −4.35398 | + | 19.0760i | ||||||
53.18 | 2.31293 | − | 1.84450i | 0 | 1.05738 | − | 4.63269i | −2.04793 | + | 1.63317i | 0 | −2.50930 | − | 10.9939i | −0.965025 | − | 2.00390i | 0 | −1.72433 | + | 7.55480i | ||||||
53.19 | 2.87082 | − | 2.28940i | 0 | 2.11016 | − | 9.24521i | 0.550556 | − | 0.439054i | 0 | 0.216354 | + | 0.947909i | −8.73539 | − | 18.1392i | 0 | 0.575376 | − | 2.52089i | ||||||
53.20 | 3.01719 | − | 2.40613i | 0 | 2.42390 | − | 10.6198i | 3.61293 | − | 2.88122i | 0 | 0.446262 | + | 1.95520i | −11.5416 | − | 23.9664i | 0 | 3.96833 | − | 17.3864i | ||||||
See next 80 embeddings (of 120 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
29.d | even | 7 | 1 | inner |
87.j | odd | 14 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 261.3.p.a | ✓ | 120 |
3.b | odd | 2 | 1 | inner | 261.3.p.a | ✓ | 120 |
29.d | even | 7 | 1 | inner | 261.3.p.a | ✓ | 120 |
87.j | odd | 14 | 1 | inner | 261.3.p.a | ✓ | 120 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
261.3.p.a | ✓ | 120 | 1.a | even | 1 | 1 | trivial |
261.3.p.a | ✓ | 120 | 3.b | odd | 2 | 1 | inner |
261.3.p.a | ✓ | 120 | 29.d | even | 7 | 1 | inner |
261.3.p.a | ✓ | 120 | 87.j | odd | 14 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(261, [\chi])\).