Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [667,2,Mod(4,667)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(667, base_ring=CyclotomicField(154))
chi = DirichletCharacter(H, H._module([28, 11]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("667.4");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 667 = 23 \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 667.u (of order \(154\), degree \(60\), 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: | \(5.32602181482\) |
Analytic rank: | \(0\) |
Dimension: | \(3480\) |
Relative dimension: | \(58\) over \(\Q(\zeta_{154})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{154}]$ |
$q$-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 | −0.549653 | − | 2.65690i | −0.0369874 | − | 0.0409665i | −4.92118 | + | 2.12720i | 2.56380 | − | 2.51202i | −0.0885136 | + | 0.120789i | 2.05343 | + | 3.05627i | 5.23925 | + | 7.46441i | 0.305159 | − | 2.98138i | −8.08340 | − | 5.43102i |
4.2 | −0.545466 | − | 2.63666i | −1.18274 | − | 1.30998i | −4.81863 | + | 2.08287i | −2.00603 | + | 1.96552i | −2.80884 | + | 3.83305i | 0.697018 | + | 1.03743i | 5.02653 | + | 7.16134i | −0.0117039 | + | 0.114346i | 6.27664 | + | 4.21710i |
4.3 | −0.531054 | − | 2.56700i | 2.03880 | + | 2.25813i | −4.47163 | + | 1.93288i | 0.250657 | − | 0.245596i | 4.71391 | − | 6.43278i | −0.956611 | − | 1.42380i | 4.32442 | + | 6.16103i | −0.636989 | + | 6.22333i | −0.763556 | − | 0.513013i |
4.4 | −0.506463 | − | 2.44813i | 0.669511 | + | 0.741536i | −3.90102 | + | 1.68623i | −0.535395 | + | 0.524583i | 1.47630 | − | 2.01461i | −0.248759 | − | 0.370247i | 3.23135 | + | 4.60374i | 0.203837 | − | 1.99147i | 1.55541 | + | 1.04504i |
4.5 | −0.504872 | − | 2.44044i | 1.39688 | + | 1.54715i | −3.86504 | + | 1.67068i | −0.360643 | + | 0.353360i | 3.07049 | − | 4.19011i | 2.29928 | + | 3.42219i | 3.16508 | + | 4.50932i | −0.136946 | + | 1.33795i | 1.04443 | + | 0.701728i |
4.6 | −0.479420 | − | 2.31741i | −1.24775 | − | 1.38198i | −3.30473 | + | 1.42848i | 1.73083 | − | 1.69588i | −2.60443 | + | 3.55410i | −0.993686 | − | 1.47898i | 2.17563 | + | 3.09964i | −0.0475267 | + | 0.464332i | −4.75985 | − | 3.19802i |
4.7 | −0.454734 | − | 2.19809i | −0.100075 | − | 0.110841i | −2.78897 | + | 1.20555i | −1.97020 | + | 1.93041i | −0.198130 | + | 0.270376i | −2.21723 | − | 3.30007i | 1.33904 | + | 1.90774i | 0.303198 | − | 2.96222i | 5.13914 | + | 3.45285i |
4.8 | −0.447110 | − | 2.16123i | −1.40317 | − | 1.55412i | −2.63519 | + | 1.13907i | 1.38554 | − | 1.35756i | −2.73145 | + | 3.72744i | 0.422913 | + | 0.629454i | 1.10417 | + | 1.57312i | −0.140941 | + | 1.37699i | −3.55349 | − | 2.38750i |
4.9 | −0.421556 | − | 2.03771i | 0.434970 | + | 0.481764i | −2.13873 | + | 0.924473i | 1.81756 | − | 1.78086i | 0.798332 | − | 1.08943i | −2.41486 | − | 3.59422i | 0.394477 | + | 0.562014i | 0.262571 | − | 2.56530i | −4.39508 | − | 2.95294i |
4.10 | −0.411927 | − | 1.99116i | −2.15255 | − | 2.38413i | −1.95921 | + | 0.846879i | −0.158097 | + | 0.154904i | −3.86049 | + | 5.26817i | 2.85919 | + | 4.25555i | 0.157018 | + | 0.223704i | −0.745094 | + | 7.27951i | 0.373563 | + | 0.250987i |
4.11 | −0.401211 | − | 1.93937i | −0.0273517 | − | 0.0302942i | −1.76434 | + | 0.762644i | −1.73901 | + | 1.70390i | −0.0477778 | + | 0.0651994i | 0.953671 | + | 1.41942i | −0.0886148 | − | 0.126250i | 0.305299 | − | 2.98275i | 4.00219 | + | 2.68897i |
4.12 | −0.376225 | − | 1.81859i | 1.56277 | + | 1.73089i | −1.32988 | + | 0.574848i | −0.702947 | + | 0.688752i | 2.55982 | − | 3.49323i | −0.807846 | − | 1.20238i | −0.588074 | − | 0.837835i | −0.248267 | + | 2.42555i | 1.51702 | + | 1.01925i |
4.13 | −0.369973 | − | 1.78837i | 0.784670 | + | 0.869084i | −1.22556 | + | 0.529753i | 2.78107 | − | 2.72491i | 1.26394 | − | 1.72482i | −0.281512 | − | 0.418995i | −0.697546 | − | 0.993800i | 0.165868 | − | 1.62052i | −5.90208 | − | 3.96545i |
4.14 | −0.346908 | − | 1.67688i | 1.92093 | + | 2.12758i | −0.855740 | + | 0.369897i | −2.77227 | + | 2.71628i | 2.90131 | − | 3.95924i | −0.980485 | − | 1.45933i | −1.05041 | − | 1.49653i | −0.531167 | + | 5.18946i | 5.51659 | + | 3.70645i |
4.15 | −0.313092 | − | 1.51342i | −1.02547 | − | 1.13579i | −0.356581 | + | 0.154134i | 0.299738 | − | 0.293685i | −1.39786 | + | 1.90758i | 0.0333896 | + | 0.0496963i | −1.43084 | − | 2.03853i | 0.0670393 | − | 0.654969i | −0.538315 | − | 0.361680i |
4.16 | −0.301495 | − | 1.45736i | 2.05489 | + | 2.27596i | −0.197172 | + | 0.0852286i | 1.60989 | − | 1.57738i | 2.69735 | − | 3.68091i | 1.68389 | + | 2.50626i | −1.52632 | − | 2.17457i | −0.651928 | + | 6.36928i | −2.78418 | − | 1.87062i |
4.17 | −0.281431 | − | 1.36038i | −0.426723 | − | 0.472629i | 0.0644155 | − | 0.0278439i | −3.04870 | + | 2.98713i | −0.522860 | + | 0.713515i | 2.40462 | + | 3.57898i | −1.65219 | − | 2.35389i | 0.264183 | − | 2.58104i | 4.92162 | + | 3.30671i |
4.18 | −0.268126 | − | 1.29606i | 0.851696 | + | 0.943321i | 0.227944 | − | 0.0985298i | 1.04828 | − | 1.02711i | 0.994242 | − | 1.35678i | 1.73800 | + | 2.58679i | −1.70954 | − | 2.43560i | 0.141000 | − | 1.37756i | −1.61227 | − | 1.08324i |
4.19 | −0.247370 | − | 1.19573i | −1.53071 | − | 1.69538i | 0.467244 | − | 0.201968i | −1.17093 | + | 1.14729i | −1.64858 | + | 2.24971i | 0.0489480 | + | 0.0728529i | −1.76008 | − | 2.50761i | −0.225783 | + | 2.20588i | 1.66150 | + | 1.11632i |
4.20 | −0.229353 | − | 1.10864i | −1.27498 | − | 1.41214i | 0.659349 | − | 0.285006i | −0.878971 | + | 0.861221i | −1.27314 | + | 1.73738i | −2.69047 | − | 4.00443i | −1.76801 | − | 2.51889i | −0.0631032 | + | 0.616513i | 1.15638 | + | 0.776941i |
See next 80 embeddings (of 3480 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
23.c | even | 11 | 1 | inner |
29.e | even | 14 | 1 | inner |
667.u | even | 154 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 667.2.u.a | ✓ | 3480 |
23.c | even | 11 | 1 | inner | 667.2.u.a | ✓ | 3480 |
29.e | even | 14 | 1 | inner | 667.2.u.a | ✓ | 3480 |
667.u | even | 154 | 1 | inner | 667.2.u.a | ✓ | 3480 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
667.2.u.a | ✓ | 3480 | 1.a | even | 1 | 1 | trivial |
667.2.u.a | ✓ | 3480 | 23.c | even | 11 | 1 | inner |
667.2.u.a | ✓ | 3480 | 29.e | even | 14 | 1 | inner |
667.2.u.a | ✓ | 3480 | 667.u | even | 154 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(667, [\chi])\).