Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [555,2,Mod(4,555)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(555, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([0, 9, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("555.4");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 555 = 3 \cdot 5 \cdot 37 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 555.bp (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: | \(4.43169731218\) |
Analytic rank: | \(0\) |
Dimension: | \(240\) |
Relative dimension: | \(40\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 | −0.483654 | + | 2.74294i | −0.984808 | + | 0.173648i | −5.41042 | − | 1.96923i | 2.17371 | − | 0.524382i | − | 2.78526i | 2.76078 | + | 3.29017i | 5.23300 | − | 9.06382i | 0.939693 | − | 0.342020i | 0.387024 | + | 6.21598i | |
4.2 | −0.482121 | + | 2.73425i | 0.984808 | − | 0.173648i | −5.36427 | − | 1.95244i | −1.84866 | − | 1.25795i | 2.77643i | −0.622659 | − | 0.742056i | 5.14824 | − | 8.91702i | 0.939693 | − | 0.342020i | 4.33083 | − | 4.44822i | ||
4.3 | −0.444573 | + | 2.52130i | −0.984808 | + | 0.173648i | −4.27990 | − | 1.55776i | −1.33654 | + | 1.79267i | − | 2.56019i | −0.953334 | − | 1.13614i | 3.27010 | − | 5.66399i | 0.939693 | − | 0.342020i | −3.92566 | − | 4.16679i | |
4.4 | −0.431485 | + | 2.44707i | 0.984808 | − | 0.173648i | −3.92261 | − | 1.42771i | 0.791716 | + | 2.09122i | 2.48482i | −2.01383 | − | 2.39999i | 2.70145 | − | 4.67904i | 0.939693 | − | 0.342020i | −5.45898 | + | 1.03506i | ||
4.5 | −0.396444 | + | 2.24835i | 0.984808 | − | 0.173648i | −3.01852 | − | 1.09865i | 1.48813 | + | 1.66898i | 2.28303i | 2.52076 | + | 3.00412i | 1.38379 | − | 2.39679i | 0.939693 | − | 0.342020i | −4.34240 | + | 2.68417i | ||
4.6 | −0.392039 | + | 2.22336i | −0.984808 | + | 0.173648i | −2.91026 | − | 1.05925i | 2.23508 | + | 0.0665805i | − | 2.25766i | −3.20631 | − | 3.82113i | 1.23837 | − | 2.14492i | 0.939693 | − | 0.342020i | −1.02427 | + | 4.94328i | |
4.7 | −0.333537 | + | 1.89158i | −0.984808 | + | 0.173648i | −1.58745 | − | 0.577784i | −2.21455 | − | 0.309466i | − | 1.92076i | 2.01071 | + | 2.39627i | −0.298365 | + | 0.516783i | 0.939693 | − | 0.342020i | 1.32401 | − | 4.08578i | |
4.8 | −0.328062 | + | 1.86053i | −0.984808 | + | 0.173648i | −1.47457 | − | 0.536701i | 0.674706 | + | 2.13185i | − | 1.88923i | 0.456273 | + | 0.543765i | −0.406933 | + | 0.704829i | 0.939693 | − | 0.342020i | −4.18772 | + | 0.555934i | |
4.9 | −0.326358 | + | 1.85087i | 0.984808 | − | 0.173648i | −1.43983 | − | 0.524054i | −2.21686 | − | 0.292450i | 1.87942i | −2.07200 | − | 2.46931i | −0.439568 | + | 0.761354i | 0.939693 | − | 0.342020i | 1.26478 | − | 4.00768i | ||
4.10 | −0.315810 | + | 1.79105i | 0.984808 | − | 0.173648i | −1.22873 | − | 0.447221i | 1.61367 | − | 1.54792i | 1.81868i | −1.35469 | − | 1.61445i | −0.629637 | + | 1.09056i | 0.939693 | − | 0.342020i | 2.26279 | + | 3.37901i | ||
4.11 | −0.277733 | + | 1.57510i | 0.984808 | − | 0.173648i | −0.524428 | − | 0.190876i | −1.34715 | − | 1.78471i | 1.59940i | 1.57453 | + | 1.87646i | −1.15310 | + | 1.99723i | 0.939693 | − | 0.342020i | 3.18525 | − | 1.62623i | ||
4.12 | −0.208975 | + | 1.18516i | 0.984808 | − | 0.173648i | 0.518457 | + | 0.188703i | 2.23588 | − | 0.0286419i | 1.20344i | 1.21615 | + | 1.44935i | −1.53543 | + | 2.65944i | 0.939693 | − | 0.342020i | −0.433299 | + | 2.65586i | ||
4.13 | −0.204840 | + | 1.16171i | −0.984808 | + | 0.173648i | 0.571779 | + | 0.208111i | 1.68974 | − | 1.46451i | − | 1.17963i | 2.65369 | + | 3.16254i | −1.53852 | + | 2.66479i | 0.939693 | − | 0.342020i | 1.35520 | + | 2.26297i | |
4.14 | −0.190715 | + | 1.08160i | −0.984808 | + | 0.173648i | 0.745904 | + | 0.271487i | 0.174198 | − | 2.22927i | − | 1.09828i | −2.17481 | − | 2.59184i | −1.53418 | + | 2.65727i | 0.939693 | − | 0.342020i | 2.37795 | + | 0.613567i | |
4.15 | −0.185254 | + | 1.05063i | −0.984808 | + | 0.173648i | 0.809888 | + | 0.294775i | −0.663270 | + | 2.13543i | − | 1.06683i | −0.415188 | − | 0.494802i | −1.52657 | + | 2.64409i | 0.939693 | − | 0.342020i | −2.12067 | − | 1.09245i | |
4.16 | −0.169137 | + | 0.959222i | 0.984808 | − | 0.173648i | 0.987886 | + | 0.359561i | −1.41362 | + | 1.73254i | 0.974019i | 0.930024 | + | 1.10836i | −1.48601 | + | 2.57384i | 0.939693 | − | 0.342020i | −1.42279 | − | 1.64901i | ||
4.17 | −0.101920 | + | 0.578017i | −0.984808 | + | 0.173648i | 1.55567 | + | 0.566218i | −2.03763 | − | 0.920904i | − | 0.586933i | −1.49229 | − | 1.77844i | −1.07277 | + | 1.85809i | 0.939693 | − | 0.342020i | 0.739973 | − | 1.08393i | |
4.18 | −0.0741078 | + | 0.420286i | 0.984808 | − | 0.173648i | 1.70824 | + | 0.621747i | 1.35836 | + | 1.77619i | 0.426770i | −2.53805 | − | 3.02473i | −0.814676 | + | 1.41106i | 0.939693 | − | 0.342020i | −0.847173 | + | 0.439272i | ||
4.19 | −0.0600890 | + | 0.340781i | −0.984808 | + | 0.173648i | 1.76686 | + | 0.643086i | 1.86567 | + | 1.23259i | − | 0.346039i | −1.02117 | − | 1.21698i | −0.671359 | + | 1.16283i | 0.939693 | − | 0.342020i | −0.532151 | + | 0.561720i | |
4.20 | −0.0425104 | + | 0.241089i | 0.984808 | − | 0.173648i | 1.82307 | + | 0.663543i | 0.433786 | − | 2.19359i | 0.244808i | −2.03953 | − | 2.43062i | −0.482280 | + | 0.835333i | 0.939693 | − | 0.342020i | 0.510409 | + | 0.197831i | ||
See next 80 embeddings (of 240 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
37.h | even | 18 | 1 | inner |
185.v | even | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 555.2.bp.a | ✓ | 240 |
5.b | even | 2 | 1 | inner | 555.2.bp.a | ✓ | 240 |
37.h | even | 18 | 1 | inner | 555.2.bp.a | ✓ | 240 |
185.v | even | 18 | 1 | inner | 555.2.bp.a | ✓ | 240 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
555.2.bp.a | ✓ | 240 | 1.a | even | 1 | 1 | trivial |
555.2.bp.a | ✓ | 240 | 5.b | even | 2 | 1 | inner |
555.2.bp.a | ✓ | 240 | 37.h | even | 18 | 1 | inner |
555.2.bp.a | ✓ | 240 | 185.v | even | 18 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(555, [\chi])\).