Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [268,2,Mod(17,268)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(268, base_ring=CyclotomicField(66))
chi = DirichletCharacter(H, H._module([0, 64]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("268.17");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 268 = 2^{2} \cdot 67 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 268.m (of order \(33\), degree \(20\), 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: | \(120\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{33})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{33}]$ |
$q$-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 | 0 | −2.63746 | + | 1.69499i | 0 | −0.492907 | + | 3.42824i | 0 | −0.397629 | − | 0.558392i | 0 | 2.83695 | − | 6.21205i | 0 | ||||||||||
17.2 | 0 | −1.67230 | + | 1.07472i | 0 | 0.298559 | − | 2.07653i | 0 | 1.54983 | + | 2.17643i | 0 | 0.395311 | − | 0.865610i | 0 | ||||||||||
17.3 | 0 | −0.0148630 | + | 0.00955186i | 0 | 0.395959 | − | 2.75395i | 0 | −2.83633 | − | 3.98307i | 0 | −1.24612 | + | 2.72861i | 0 | ||||||||||
17.4 | 0 | 0.505110 | − | 0.324615i | 0 | −0.422041 | + | 2.93536i | 0 | −0.322193 | − | 0.452456i | 0 | −1.09648 | + | 2.40096i | 0 | ||||||||||
17.5 | 0 | 1.68771 | − | 1.08462i | 0 | 0.308909 | − | 2.14851i | 0 | 2.06618 | + | 2.90154i | 0 | 0.425703 | − | 0.932160i | 0 | ||||||||||
17.6 | 0 | 2.67078 | − | 1.71641i | 0 | −0.167112 | + | 1.16229i | 0 | −1.95872 | − | 2.75064i | 0 | 2.94078 | − | 6.43941i | 0 | ||||||||||
21.1 | 0 | −1.13773 | + | 2.49127i | 0 | −3.83671 | − | 1.12656i | 0 | 1.55028 | − | 4.47923i | 0 | −2.94743 | − | 3.40151i | 0 | ||||||||||
21.2 | 0 | −1.02902 | + | 2.25325i | 0 | 2.64873 | + | 0.777737i | 0 | −0.432784 | + | 1.25045i | 0 | −2.05366 | − | 2.37005i | 0 | ||||||||||
21.3 | 0 | −0.160887 | + | 0.352293i | 0 | −1.66715 | − | 0.489520i | 0 | −1.05867 | + | 3.05882i | 0 | 1.86636 | + | 2.15389i | 0 | ||||||||||
21.4 | 0 | 0.368818 | − | 0.807599i | 0 | −0.0737807 | − | 0.0216640i | 0 | 0.660618 | − | 1.90873i | 0 | 1.44839 | + | 1.67153i | 0 | ||||||||||
21.5 | 0 | 0.796606 | − | 1.74432i | 0 | 3.57137 | + | 1.04865i | 0 | 0.0733205 | − | 0.211846i | 0 | −0.443504 | − | 0.511831i | 0 | ||||||||||
21.6 | 0 | 1.38323 | − | 3.02886i | 0 | −2.47014 | − | 0.725297i | 0 | −0.0923903 | + | 0.266944i | 0 | −5.29607 | − | 6.11199i | 0 | ||||||||||
33.1 | 0 | −2.99106 | + | 0.878254i | 0 | −0.544051 | − | 0.627868i | 0 | −0.623535 | − | 0.321455i | 0 | 5.65133 | − | 3.63189i | 0 | ||||||||||
33.2 | 0 | −0.748191 | + | 0.219689i | 0 | −1.58768 | − | 1.83229i | 0 | −1.29012 | − | 0.665104i | 0 | −2.01223 | + | 1.29318i | 0 | ||||||||||
33.3 | 0 | −0.680838 | + | 0.199912i | 0 | 2.09126 | + | 2.41344i | 0 | −0.652698 | − | 0.336489i | 0 | −2.10019 | + | 1.34971i | 0 | ||||||||||
33.4 | 0 | 1.26064 | − | 0.370157i | 0 | 0.454892 | + | 0.524973i | 0 | 3.21543 | + | 1.65767i | 0 | −1.07157 | + | 0.688655i | 0 | ||||||||||
33.5 | 0 | 2.12465 | − | 0.623854i | 0 | −1.97351 | − | 2.27755i | 0 | 0.0956759 | + | 0.0493244i | 0 | 1.60119 | − | 1.02903i | 0 | ||||||||||
33.6 | 0 | 3.19808 | − | 0.939041i | 0 | 1.46527 | + | 1.69102i | 0 | −3.25284 | − | 1.67696i | 0 | 6.82216 | − | 4.38434i | 0 | ||||||||||
49.1 | 0 | −2.26129 | − | 1.45324i | 0 | 0.354018 | + | 2.46225i | 0 | −0.314804 | − | 0.0300601i | 0 | 1.75526 | + | 3.84348i | 0 | ||||||||||
49.2 | 0 | −1.35538 | − | 0.871048i | 0 | −0.136981 | − | 0.952723i | 0 | 1.40117 | + | 0.133795i | 0 | −0.167923 | − | 0.367700i | 0 | ||||||||||
See next 80 embeddings (of 120 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
67.g | even | 33 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 268.2.m.a | ✓ | 120 |
67.g | even | 33 | 1 | inner | 268.2.m.a | ✓ | 120 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
268.2.m.a | ✓ | 120 | 1.a | even | 1 | 1 | trivial |
268.2.m.a | ✓ | 120 | 67.g | even | 33 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(268, [\chi])\).