Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [273,2,Mod(209,273)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(273, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("273.209");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 273 = 3 \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 273.e (of order \(2\), degree \(1\), 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: | \(2.17991597518\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
209.1 | − | 2.62996i | −0.518189 | + | 1.65272i | −4.91669 | 2.48925 | 4.34658 | + | 1.36282i | 1.33971 | − | 2.28149i | 7.67077i | −2.46296 | − | 1.71284i | − | 6.54663i | ||||||||
209.2 | − | 2.62996i | 0.518189 | − | 1.65272i | −4.91669 | −2.48925 | −4.34658 | − | 1.36282i | 1.33971 | + | 2.28149i | 7.67077i | −2.46296 | − | 1.71284i | 6.54663i | |||||||||
209.3 | − | 2.37766i | −1.58497 | + | 0.698482i | −3.65324 | −1.30604 | 1.66075 | + | 3.76851i | −0.0678621 | + | 2.64488i | 3.93085i | 2.02425 | − | 2.21414i | 3.10530i | |||||||||
209.4 | − | 2.37766i | 1.58497 | − | 0.698482i | −3.65324 | 1.30604 | −1.66075 | − | 3.76851i | −0.0678621 | − | 2.64488i | 3.93085i | 2.02425 | − | 2.21414i | − | 3.10530i | ||||||||
209.5 | − | 1.95979i | −1.02881 | − | 1.39340i | −1.84076 | −2.34217 | −2.73076 | + | 2.01624i | 0.446056 | − | 2.60788i | − | 0.312071i | −0.883116 | + | 2.86707i | 4.59015i | ||||||||
209.6 | − | 1.95979i | 1.02881 | + | 1.39340i | −1.84076 | 2.34217 | 2.73076 | − | 2.01624i | 0.446056 | + | 2.60788i | − | 0.312071i | −0.883116 | + | 2.86707i | − | 4.59015i | |||||||
209.7 | − | 1.86253i | −0.150306 | + | 1.72552i | −1.46902 | −2.96439 | 3.21383 | + | 0.279950i | −2.41066 | − | 1.09028i | − | 0.988960i | −2.95482 | − | 0.518713i | 5.52126i | ||||||||
209.8 | − | 1.86253i | 0.150306 | − | 1.72552i | −1.46902 | 2.96439 | −3.21383 | − | 0.279950i | −2.41066 | + | 1.09028i | − | 0.988960i | −2.95482 | − | 0.518713i | − | 5.52126i | |||||||
209.9 | − | 1.71574i | −1.66803 | − | 0.466569i | −0.943779 | 2.59497 | −0.800512 | + | 2.86191i | 2.59496 | + | 0.515930i | − | 1.81221i | 2.56463 | + | 1.55650i | − | 4.45230i | |||||||
209.10 | − | 1.71574i | 1.66803 | + | 0.466569i | −0.943779 | −2.59497 | 0.800512 | − | 2.86191i | 2.59496 | − | 0.515930i | − | 1.81221i | 2.56463 | + | 1.55650i | 4.45230i | ||||||||
209.11 | − | 0.875536i | −1.71925 | + | 0.210227i | 1.23344 | −0.171607 | 0.184061 | + | 1.50526i | −2.26296 | − | 1.37077i | − | 2.83099i | 2.91161 | − | 0.722864i | 0.150248i | ||||||||
209.12 | − | 0.875536i | 1.71925 | − | 0.210227i | 1.23344 | 0.171607 | −0.184061 | − | 1.50526i | −2.26296 | + | 1.37077i | − | 2.83099i | 2.91161 | − | 0.722864i | − | 0.150248i | |||||||
209.13 | − | 0.633614i | −0.915380 | + | 1.47040i | 1.59853 | 0.119728 | 0.931666 | + | 0.579998i | 2.16525 | − | 1.52042i | − | 2.28008i | −1.32416 | − | 2.69195i | − | 0.0758615i | |||||||
209.14 | − | 0.633614i | 0.915380 | − | 1.47040i | 1.59853 | −0.119728 | −0.931666 | − | 0.579998i | 2.16525 | + | 1.52042i | − | 2.28008i | −1.32416 | − | 2.69195i | 0.0758615i | ||||||||
209.15 | − | 0.0920595i | −0.749855 | + | 1.56132i | 1.99153 | 3.32369 | 0.143734 | + | 0.0690313i | −0.804490 | + | 2.52048i | − | 0.367458i | −1.87543 | − | 2.34153i | − | 0.305977i | |||||||
209.16 | − | 0.0920595i | 0.749855 | − | 1.56132i | 1.99153 | −3.32369 | −0.143734 | − | 0.0690313i | −0.804490 | − | 2.52048i | − | 0.367458i | −1.87543 | − | 2.34153i | 0.305977i | ||||||||
209.17 | 0.0920595i | −0.749855 | − | 1.56132i | 1.99153 | 3.32369 | 0.143734 | − | 0.0690313i | −0.804490 | − | 2.52048i | 0.367458i | −1.87543 | + | 2.34153i | 0.305977i | ||||||||||
209.18 | 0.0920595i | 0.749855 | + | 1.56132i | 1.99153 | −3.32369 | −0.143734 | + | 0.0690313i | −0.804490 | + | 2.52048i | 0.367458i | −1.87543 | + | 2.34153i | − | 0.305977i | |||||||||
209.19 | 0.633614i | −0.915380 | − | 1.47040i | 1.59853 | 0.119728 | 0.931666 | − | 0.579998i | 2.16525 | + | 1.52042i | 2.28008i | −1.32416 | + | 2.69195i | 0.0758615i | ||||||||||
209.20 | 0.633614i | 0.915380 | + | 1.47040i | 1.59853 | −0.119728 | −0.931666 | + | 0.579998i | 2.16525 | − | 1.52042i | 2.28008i | −1.32416 | + | 2.69195i | − | 0.0758615i | |||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
21.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 273.2.e.a | ✓ | 32 |
3.b | odd | 2 | 1 | inner | 273.2.e.a | ✓ | 32 |
7.b | odd | 2 | 1 | inner | 273.2.e.a | ✓ | 32 |
21.c | even | 2 | 1 | inner | 273.2.e.a | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
273.2.e.a | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
273.2.e.a | ✓ | 32 | 3.b | odd | 2 | 1 | inner |
273.2.e.a | ✓ | 32 | 7.b | odd | 2 | 1 | inner |
273.2.e.a | ✓ | 32 | 21.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(273, [\chi])\).