Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [138,3,Mod(29,138)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(138, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([11, 18]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("138.29");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 138 = 2 \cdot 3 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 138.g (of order \(22\), degree \(10\), 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: | \(3.76022764817\) |
Analytic rank: | \(0\) |
Dimension: | \(160\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{22})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{22}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
29.1 | −1.28641 | − | 0.587486i | −2.98078 | − | 0.339083i | 1.30972 | + | 1.51150i | −3.03728 | − | 4.72610i | 3.63531 | + | 2.18736i | −0.130023 | + | 0.904330i | −0.796860 | − | 2.71386i | 8.77005 | + | 2.02146i | 1.13068 | + | 7.86408i |
29.2 | −1.28641 | − | 0.587486i | −2.45333 | − | 1.72660i | 1.30972 | + | 1.51150i | 2.91988 | + | 4.54342i | 2.14165 | + | 3.66242i | 1.52608 | − | 10.6142i | −0.796860 | − | 2.71386i | 3.03768 | + | 8.47186i | −1.08698 | − | 7.56011i |
29.3 | −1.28641 | − | 0.587486i | −0.901649 | − | 2.86130i | 1.30972 | + | 1.51150i | 1.85466 | + | 2.88591i | −0.521077 | + | 4.21052i | −1.93124 | + | 13.4321i | −0.796860 | − | 2.71386i | −7.37406 | + | 5.15978i | −0.690432 | − | 4.80206i |
29.4 | −1.28641 | − | 0.587486i | −0.671287 | + | 2.92393i | 1.30972 | + | 1.51150i | 3.78542 | + | 5.89022i | 2.58132 | − | 3.36701i | 0.377062 | − | 2.62252i | −0.796860 | − | 2.71386i | −8.09875 | − | 3.92560i | −1.40919 | − | 9.80114i |
29.5 | −1.28641 | − | 0.587486i | −0.200212 | + | 2.99331i | 1.30972 | + | 1.51150i | −4.05813 | − | 6.31458i | 2.01608 | − | 3.73302i | −0.756114 | + | 5.25889i | −0.796860 | − | 2.71386i | −8.91983 | − | 1.19859i | 1.51072 | + | 10.5073i |
29.6 | −1.28641 | − | 0.587486i | 1.85221 | − | 2.35994i | 1.30972 | + | 1.51150i | −1.93323 | − | 3.00816i | −3.76914 | + | 1.94772i | 0.370720 | − | 2.57842i | −0.796860 | − | 2.71386i | −2.13865 | − | 8.74221i | 0.719680 | + | 5.00549i |
29.7 | −1.28641 | − | 0.587486i | 2.39931 | + | 1.80092i | 1.30972 | + | 1.51150i | −1.08843 | − | 1.69363i | −2.02849 | − | 3.72629i | 1.71626 | − | 11.9369i | −0.796860 | − | 2.71386i | 2.51336 | + | 8.64193i | 0.405187 | + | 2.81814i |
29.8 | −1.28641 | − | 0.587486i | 2.89781 | + | 0.776326i | 1.30972 | + | 1.51150i | 1.55712 | + | 2.42292i | −3.27171 | − | 2.70110i | −1.17275 | + | 8.15664i | −0.796860 | − | 2.71386i | 7.79464 | + | 4.49930i | −0.579665 | − | 4.03166i |
29.9 | 1.28641 | + | 0.587486i | −2.99735 | + | 0.126139i | 1.30972 | + | 1.51150i | −2.91988 | − | 4.54342i | −3.92993 | − | 1.59863i | 1.52608 | − | 10.6142i | 0.796860 | + | 2.71386i | 8.96818 | − | 0.756167i | −1.08698 | − | 7.56011i |
29.10 | 1.28641 | + | 0.587486i | −2.69091 | − | 1.32627i | 1.30972 | + | 1.51150i | 3.03728 | + | 4.72610i | −2.68246 | − | 3.28701i | −0.130023 | + | 0.904330i | 0.796860 | + | 2.71386i | 5.48199 | + | 7.13777i | 1.13068 | + | 7.86408i |
29.11 | 1.28641 | + | 0.587486i | −2.30545 | + | 1.91961i | 1.30972 | + | 1.51150i | −1.85466 | − | 2.88591i | −4.09351 | + | 1.11499i | −1.93124 | + | 13.4321i | 0.796860 | + | 2.71386i | 1.63020 | − | 8.85113i | −0.690432 | − | 4.80206i |
29.12 | 1.28641 | + | 0.587486i | 0.282296 | + | 2.98669i | 1.30972 | + | 1.51150i | 1.93323 | + | 3.00816i | −1.39149 | + | 4.00796i | 0.370720 | − | 2.57842i | 0.796860 | + | 2.71386i | −8.84062 | + | 1.68626i | 0.719680 | + | 5.00549i |
29.13 | 1.28641 | + | 0.587486i | 1.01607 | − | 2.82269i | 1.30972 | + | 1.51150i | −3.78542 | − | 5.89022i | 2.96538 | − | 3.03422i | 0.377062 | − | 2.62252i | 0.796860 | + | 2.71386i | −6.93519 | − | 5.73613i | −1.40919 | − | 9.80114i |
29.14 | 1.28641 | + | 0.587486i | 1.44988 | − | 2.62638i | 1.30972 | + | 1.51150i | 4.05813 | + | 6.31458i | 3.40810 | − | 2.52683i | −0.756114 | + | 5.25889i | 0.796860 | + | 2.71386i | −4.79571 | − | 7.61585i | 1.51072 | + | 10.5073i |
29.15 | 1.28641 | + | 0.587486i | 2.85751 | + | 0.913589i | 1.30972 | + | 1.51150i | −1.55712 | − | 2.42292i | 3.13922 | + | 2.85400i | −1.17275 | + | 8.15664i | 0.796860 | + | 2.71386i | 7.33071 | + | 5.22117i | −0.579665 | − | 4.03166i |
29.16 | 1.28641 | + | 0.587486i | 2.99208 | − | 0.217869i | 1.30972 | + | 1.51150i | 1.08843 | + | 1.69363i | 3.97705 | + | 1.47753i | 1.71626 | − | 11.9369i | 0.796860 | + | 2.71386i | 8.90507 | − | 1.30376i | 0.405187 | + | 2.81814i |
35.1 | −0.764582 | − | 1.18971i | −2.91647 | + | 0.703009i | −0.830830 | + | 1.81926i | −0.00374340 | − | 0.0127488i | 3.06625 | + | 2.93225i | 1.68160 | + | 1.94067i | 2.79964 | − | 0.402527i | 8.01156 | − | 4.10060i | −0.0123053 | + | 0.0142011i |
35.2 | −0.764582 | − | 1.18971i | −1.64224 | − | 2.51059i | −0.830830 | + | 1.81926i | −2.24222 | − | 7.63629i | −1.73126 | + | 3.87334i | 0.444889 | + | 0.513429i | 2.79964 | − | 0.402527i | −3.60612 | + | 8.24596i | −7.37063 | + | 8.50616i |
35.3 | −0.764582 | − | 1.18971i | −0.894326 | + | 2.86360i | −0.830830 | + | 1.81926i | −0.674627 | − | 2.29757i | 4.09064 | − | 1.12546i | 1.86579 | + | 2.15324i | 2.79964 | − | 0.402527i | −7.40036 | − | 5.12197i | −2.21764 | + | 2.55929i |
35.4 | −0.764582 | − | 1.18971i | −0.618173 | − | 2.93562i | −0.830830 | + | 1.81926i | 2.12160 | + | 7.22551i | −3.01990 | + | 2.97997i | 7.03927 | + | 8.12375i | 2.79964 | − | 0.402527i | −8.23573 | + | 3.62944i | 6.97414 | − | 8.04859i |
See next 80 embeddings (of 160 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
23.c | even | 11 | 1 | inner |
69.h | odd | 22 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 138.3.g.a | ✓ | 160 |
3.b | odd | 2 | 1 | inner | 138.3.g.a | ✓ | 160 |
23.c | even | 11 | 1 | inner | 138.3.g.a | ✓ | 160 |
69.h | odd | 22 | 1 | inner | 138.3.g.a | ✓ | 160 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
138.3.g.a | ✓ | 160 | 1.a | even | 1 | 1 | trivial |
138.3.g.a | ✓ | 160 | 3.b | odd | 2 | 1 | inner |
138.3.g.a | ✓ | 160 | 23.c | even | 11 | 1 | inner |
138.3.g.a | ✓ | 160 | 69.h | odd | 22 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(138, [\chi])\).