Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [268,2,Mod(231,268)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(268, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("268.231");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 268 = 2^{2} \cdot 67 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 268.h (of order \(6\), 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: | \(2.13999077417\) |
Analytic rank: | \(0\) |
Dimension: | \(64\) |
Relative dimension: | \(32\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
231.1 | −1.40647 | + | 0.147767i | 0.455030 | 1.95633 | − | 0.415660i | − | 2.16145i | −0.639988 | + | 0.0672385i | 0.987783 | − | 1.71089i | −2.69010 | + | 0.873696i | −2.79295 | 0.319391 | + | 3.04002i | |||||
231.2 | −1.38407 | + | 0.290422i | 0.541462 | 1.83131 | − | 0.803930i | 1.10815i | −0.749423 | + | 0.157253i | −1.59264 | + | 2.75853i | −2.30119 | + | 1.64455i | −2.70682 | −0.321830 | − | 1.53375i | ||||||
231.3 | −1.37412 | − | 0.334372i | −2.28807 | 1.77639 | + | 0.918932i | 2.12424i | 3.14407 | + | 0.765066i | 2.00882 | − | 3.47937i | −2.13370 | − | 1.85670i | 2.23525 | 0.710288 | − | 2.91896i | ||||||
231.4 | −1.31834 | − | 0.511846i | −2.39012 | 1.47603 | + | 1.34957i | − | 2.15536i | 3.15098 | + | 1.22337i | −2.12266 | + | 3.67656i | −1.25513 | − | 2.53469i | 2.71266 | −1.10321 | + | 2.84149i | |||||
231.5 | −1.30208 | + | 0.551888i | 2.91499 | 1.39084 | − | 1.43721i | 3.25556i | −3.79555 | + | 1.60875i | 1.81038 | − | 3.13566i | −1.01781 | + | 2.63895i | 5.49714 | −1.79671 | − | 4.23901i | ||||||
231.6 | −1.27284 | − | 0.616344i | 3.05470 | 1.24024 | + | 1.56901i | − | 0.378054i | −3.88814 | − | 1.88275i | −1.13094 | + | 1.95884i | −0.611571 | − | 2.76152i | 6.33119 | −0.233012 | + | 0.481202i | |||||
231.7 | −1.22124 | − | 0.713141i | 0.817393 | 0.982860 | + | 1.74183i | − | 3.21972i | −0.998234 | − | 0.582916i | 1.30581 | − | 2.26173i | 0.0418643 | − | 2.82812i | −2.33187 | −2.29611 | + | 3.93205i | |||||
231.8 | −1.12899 | + | 0.851692i | −2.91499 | 0.549240 | − | 1.92311i | 3.25556i | 3.29099 | − | 2.48267i | −1.81038 | + | 3.13566i | 1.01781 | + | 2.63895i | 5.49714 | −2.77274 | − | 3.67550i | ||||||
231.9 | −0.955026 | − | 1.04304i | 1.04429 | −0.175852 | + | 1.99225i | 3.14744i | −0.997321 | − | 1.08923i | −0.0992949 | + | 0.171984i | 2.24594 | − | 1.71923i | −1.90946 | 3.28289 | − | 3.00588i | ||||||
231.10 | −0.943549 | + | 1.05343i | −0.541462 | −0.219431 | − | 1.98793i | 1.10815i | 0.510896 | − | 0.570393i | 1.59264 | − | 2.75853i | 2.30119 | + | 1.64455i | −2.70682 | −1.16735 | − | 1.04559i | ||||||
231.11 | −0.831206 | + | 1.14416i | −0.455030 | −0.618192 | − | 1.90206i | − | 2.16145i | 0.378224 | − | 0.520626i | −0.987783 | + | 1.71089i | 2.69010 | + | 0.873696i | −2.79295 | 2.47304 | + | 1.79661i | |||||
231.12 | −0.730066 | − | 1.21120i | −1.87421 | −0.934007 | + | 1.76851i | 1.10069i | 1.36830 | + | 2.27004i | 0.173063 | − | 0.299754i | 2.82391 | − | 0.159861i | 0.512664 | 1.33315 | − | 0.803575i | ||||||
231.13 | −0.397483 | + | 1.35721i | 2.28807 | −1.68401 | − | 1.07893i | 2.12424i | −0.909469 | + | 3.10538i | −2.00882 | + | 3.47937i | 2.13370 | − | 1.85670i | 2.23525 | −2.88304 | − | 0.844351i | ||||||
231.14 | −0.327458 | − | 1.37578i | 1.92148 | −1.78554 | + | 0.901020i | − | 0.452542i | −0.629203 | − | 2.64353i | 1.25960 | − | 2.18169i | 1.82430 | + | 2.16147i | 0.692079 | −0.622599 | + | 0.148189i | |||||
231.15 | −0.215897 | + | 1.39764i | 2.39012 | −1.90678 | − | 0.603491i | − | 2.15536i | −0.516019 | + | 3.34052i | 2.12266 | − | 3.67656i | 1.25513 | − | 2.53469i | 2.71266 | 3.01241 | + | 0.465335i | |||||
231.16 | −0.193483 | − | 1.40092i | −2.17140 | −1.92513 | + | 0.542107i | − | 4.02184i | 0.420130 | + | 3.04195i | 0.863606 | − | 1.49581i | 1.13193 | + | 2.59205i | 1.71500 | −5.63426 | + | 0.778158i | |||||
231.17 | −0.102650 | + | 1.41048i | −3.05470 | −1.97893 | − | 0.289571i | − | 0.378054i | 0.313564 | − | 4.30860i | 1.13094 | − | 1.95884i | 0.611571 | − | 2.76152i | 6.33119 | 0.533239 | + | 0.0388071i | |||||
231.18 | 0.00697761 | + | 1.41420i | −0.817393 | −1.99990 | + | 0.0197354i | − | 3.21972i | −0.00570345 | − | 1.15595i | −1.30581 | + | 2.26173i | −0.0418643 | − | 2.82812i | −2.33187 | 4.55331 | − | 0.0224659i | |||||
231.19 | 0.165823 | − | 1.40446i | −0.657653 | −1.94501 | − | 0.465783i | 1.64739i | −0.109054 | + | 0.923646i | −1.75445 | + | 3.03879i | −0.976699 | + | 2.65444i | −2.56749 | 2.31369 | + | 0.273175i | ||||||
231.20 | 0.425784 | + | 1.34859i | −1.04429 | −1.63742 | + | 1.14842i | 3.14744i | −0.444640 | − | 1.40832i | 0.0992949 | − | 0.171984i | −2.24594 | − | 1.71923i | −1.90946 | −4.24462 | + | 1.34013i | ||||||
See all 64 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
67.d | odd | 6 | 1 | inner |
268.h | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 268.2.h.a | ✓ | 64 |
4.b | odd | 2 | 1 | inner | 268.2.h.a | ✓ | 64 |
67.d | odd | 6 | 1 | inner | 268.2.h.a | ✓ | 64 |
268.h | even | 6 | 1 | inner | 268.2.h.a | ✓ | 64 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
268.2.h.a | ✓ | 64 | 1.a | even | 1 | 1 | trivial |
268.2.h.a | ✓ | 64 | 4.b | odd | 2 | 1 | inner |
268.2.h.a | ✓ | 64 | 67.d | odd | 6 | 1 | inner |
268.2.h.a | ✓ | 64 | 268.h | even | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(268, [\chi])\).