Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [228,2,Mod(23,228)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(228, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([9, 9, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("228.23");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 228 = 2^{2} \cdot 3 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 228.v (of order \(18\), 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: | \(1.82058916609\) |
Analytic rank: | \(0\) |
Dimension: | \(216\) |
Relative dimension: | \(36\) over \(\Q(\zeta_{18})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{18}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
23.1 | −1.41413 | − | 0.0157672i | 0.174671 | − | 1.72322i | 1.99950 | + | 0.0445937i | −0.894860 | − | 1.06645i | −0.274177 | + | 2.43410i | −3.30990 | − | 1.91097i | −2.82685 | − | 0.0945877i | −2.93898 | − | 0.601994i | 1.24863 | + | 1.52221i |
23.2 | −1.40660 | − | 0.146591i | 0.0483859 | + | 1.73137i | 1.95702 | + | 0.412388i | 2.52711 | + | 3.01169i | 0.185744 | − | 2.44244i | 1.60040 | + | 0.923992i | −2.69229 | − | 0.866945i | −2.99532 | + | 0.167548i | −3.11313 | − | 4.60668i |
23.3 | −1.39258 | − | 0.246404i | −1.60968 | + | 0.639475i | 1.87857 | + | 0.686276i | −0.942247 | − | 1.12293i | 2.39918 | − | 0.493889i | 0.536440 | + | 0.309714i | −2.44696 | − | 1.41858i | 2.18214 | − | 2.05870i | 1.03546 | + | 1.79594i |
23.4 | −1.32091 | + | 0.505162i | −1.37188 | − | 1.05734i | 1.48962 | − | 1.33455i | −0.118249 | − | 0.140923i | 2.34625 | + | 0.703629i | 1.88835 | + | 1.09024i | −1.29350 | + | 2.51532i | 0.764084 | + | 2.90106i | 0.227385 | + | 0.126413i |
23.5 | −1.22517 | − | 0.706378i | 1.73132 | + | 0.0503667i | 1.00206 | + | 1.73086i | 0.942247 | + | 1.12293i | −2.08557 | − | 1.28467i | −0.536440 | − | 0.309714i | −0.00504802 | − | 2.82842i | 2.99493 | + | 0.174402i | −0.361198 | − | 2.04135i |
23.6 | −1.18680 | + | 0.769092i | 1.18231 | − | 1.26576i | 0.816994 | − | 1.82552i | 2.15444 | + | 2.56756i | −0.429685 | + | 2.41151i | 0.213070 | + | 0.123016i | 0.434383 | + | 2.79487i | −0.204279 | − | 2.99304i | −4.53159 | − | 1.39022i |
23.7 | −1.17174 | − | 0.791847i | 0.546697 | + | 1.64351i | 0.745957 | + | 1.85568i | −2.52711 | − | 3.01169i | 0.660820 | − | 2.35867i | −1.60040 | − | 0.923992i | 0.595347 | − | 2.76506i | −2.40224 | + | 1.79700i | 0.576319 | + | 5.53000i |
23.8 | −1.16179 | + | 0.806381i | 0.524795 | + | 1.65063i | 0.699499 | − | 1.87369i | −1.25702 | − | 1.49806i | −1.94074 | − | 1.49450i | 2.57863 | + | 1.48877i | 0.698237 | + | 2.74089i | −2.44918 | + | 1.73249i | 2.66839 | + | 0.726786i |
23.9 | −1.09342 | − | 0.896904i | −0.753513 | − | 1.55956i | 0.391126 | + | 1.96138i | 0.894860 | + | 1.06645i | −0.574868 | + | 2.38108i | 3.30990 | + | 1.91097i | 1.33151 | − | 2.49541i | −1.86444 | + | 2.35029i | −0.0219504 | − | 1.96868i |
23.10 | −1.08345 | + | 0.908925i | −1.12016 | + | 1.32107i | 0.347710 | − | 1.96954i | 0.218260 | + | 0.260112i | 0.0128810 | − | 2.44946i | −3.52551 | − | 2.03545i | 1.41344 | + | 2.44993i | −0.490465 | − | 2.95964i | −0.472896 | − | 0.0834354i |
23.11 | −0.709055 | + | 1.22362i | 1.68017 | − | 0.420760i | −0.994483 | − | 1.73522i | −2.09742 | − | 2.49961i | −0.676481 | + | 2.35422i | −2.63219 | − | 1.51969i | 2.82839 | + | 0.0135021i | 2.64592 | − | 1.41389i | 4.54576 | − | 0.794084i |
23.12 | −0.687167 | − | 1.23604i | 0.927511 | − | 1.46278i | −1.05560 | + | 1.69873i | 0.118249 | + | 0.140923i | −2.44541 | − | 0.141271i | −1.88835 | − | 1.09024i | 2.82509 | + | 0.137458i | −1.27945 | − | 2.71349i | 0.0929307 | − | 0.242998i |
23.13 | −0.557050 | + | 1.29988i | −1.38187 | − | 1.04424i | −1.37939 | − | 1.44820i | 1.43000 | + | 1.70421i | 2.12716 | − | 1.21457i | −1.04816 | − | 0.605157i | 2.65088 | − | 0.986326i | 0.819119 | + | 2.88601i | −3.01185 | + | 0.909504i |
23.14 | −0.414780 | − | 1.35202i | −1.54392 | − | 0.785048i | −1.65592 | + | 1.12158i | −2.15444 | − | 2.56756i | −0.421012 | + | 2.41304i | −0.213070 | − | 0.123016i | 2.20324 | + | 1.77362i | 1.76740 | + | 2.42411i | −2.57778 | + | 3.97782i |
23.15 | −0.408823 | + | 1.35383i | 1.38187 | + | 1.04424i | −1.66573 | − | 1.10696i | 1.43000 | + | 1.70421i | −1.97867 | + | 1.44391i | 1.04816 | + | 0.605157i | 2.17962 | − | 1.80257i | 0.819119 | + | 2.88601i | −2.89183 | + | 1.23926i |
23.16 | −0.371649 | − | 1.36451i | 0.0714037 | + | 1.73058i | −1.72375 | + | 1.01423i | 1.25702 | + | 1.49806i | 2.33485 | − | 0.740598i | −2.57863 | − | 1.48877i | 2.02456 | + | 1.97513i | −2.98980 | + | 0.247139i | 1.57694 | − | 2.27196i |
23.17 | −0.245722 | − | 1.39270i | 1.50444 | + | 0.858283i | −1.87924 | + | 0.684435i | −0.218260 | − | 0.260112i | 0.825659 | − | 2.30614i | 3.52551 | + | 2.03545i | 1.41498 | + | 2.44904i | 1.52670 | + | 2.58248i | −0.308628 | + | 0.367887i |
23.18 | −0.243359 | + | 1.39312i | −1.68017 | + | 0.420760i | −1.88155 | − | 0.678056i | −2.09742 | − | 2.49961i | −0.177284 | − | 2.44307i | 2.63219 | + | 1.51969i | 1.40250 | − | 2.45621i | 2.64592 | − | 1.41389i | 3.99268 | − | 2.31366i |
23.19 | 0.243359 | − | 1.39312i | −1.72275 | + | 0.179266i | −1.88155 | − | 0.678056i | 2.09742 | + | 2.49961i | −0.169508 | + | 2.44362i | 2.63219 | + | 1.51969i | −1.40250 | + | 2.45621i | 2.93573 | − | 0.617661i | 3.99268 | − | 2.31366i |
23.20 | 0.245722 | + | 1.39270i | 1.12016 | − | 1.32107i | −1.87924 | + | 0.684435i | 0.218260 | + | 0.260112i | 2.11511 | + | 1.23544i | 3.52551 | + | 2.03545i | −1.41498 | − | 2.44904i | −0.490465 | − | 2.95964i | −0.308628 | + | 0.367887i |
See next 80 embeddings (of 216 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
4.b | odd | 2 | 1 | inner |
12.b | even | 2 | 1 | inner |
19.e | even | 9 | 1 | inner |
57.l | odd | 18 | 1 | inner |
76.l | odd | 18 | 1 | inner |
228.v | even | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 228.2.v.a | ✓ | 216 |
3.b | odd | 2 | 1 | inner | 228.2.v.a | ✓ | 216 |
4.b | odd | 2 | 1 | inner | 228.2.v.a | ✓ | 216 |
12.b | even | 2 | 1 | inner | 228.2.v.a | ✓ | 216 |
19.e | even | 9 | 1 | inner | 228.2.v.a | ✓ | 216 |
57.l | odd | 18 | 1 | inner | 228.2.v.a | ✓ | 216 |
76.l | odd | 18 | 1 | inner | 228.2.v.a | ✓ | 216 |
228.v | even | 18 | 1 | inner | 228.2.v.a | ✓ | 216 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
228.2.v.a | ✓ | 216 | 1.a | even | 1 | 1 | trivial |
228.2.v.a | ✓ | 216 | 3.b | odd | 2 | 1 | inner |
228.2.v.a | ✓ | 216 | 4.b | odd | 2 | 1 | inner |
228.2.v.a | ✓ | 216 | 12.b | even | 2 | 1 | inner |
228.2.v.a | ✓ | 216 | 19.e | even | 9 | 1 | inner |
228.2.v.a | ✓ | 216 | 57.l | odd | 18 | 1 | inner |
228.2.v.a | ✓ | 216 | 76.l | odd | 18 | 1 | inner |
228.2.v.a | ✓ | 216 | 228.v | even | 18 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(228, [\chi])\).