Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [136,3,Mod(35,136)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(136, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("136.35");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 136 = 2^{3} \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 136.f (of order \(2\), degree \(1\), 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: | \(3.70573159530\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
35.1 | −1.89539 | − | 0.638369i | 0.544852 | 3.18497 | + | 2.41991i | 9.70934i | −1.03270 | − | 0.347816i | − | 5.41501i | −4.49195 | − | 6.61985i | −8.70314 | 6.19813 | − | 18.4029i | |||||||
35.2 | −1.89539 | + | 0.638369i | 0.544852 | 3.18497 | − | 2.41991i | − | 9.70934i | −1.03270 | + | 0.347816i | 5.41501i | −4.49195 | + | 6.61985i | −8.70314 | 6.19813 | + | 18.4029i | |||||||
35.3 | −1.86718 | − | 0.716676i | 4.09644 | 2.97275 | + | 2.67633i | − | 4.29268i | −7.64881 | − | 2.93582i | − | 5.81927i | −3.63261 | − | 7.12770i | 7.78083 | −3.07646 | + | 8.01523i | ||||||
35.4 | −1.86718 | + | 0.716676i | 4.09644 | 2.97275 | − | 2.67633i | 4.29268i | −7.64881 | + | 2.93582i | 5.81927i | −3.63261 | + | 7.12770i | 7.78083 | −3.07646 | − | 8.01523i | ||||||||
35.5 | −1.73617 | − | 0.992835i | −5.08000 | 2.02856 | + | 3.44746i | 1.34644i | 8.81974 | + | 5.04361i | 6.95342i | −0.0991579 | − | 7.99939i | 16.8064 | 1.33679 | − | 2.33764i | ||||||||
35.6 | −1.73617 | + | 0.992835i | −5.08000 | 2.02856 | − | 3.44746i | − | 1.34644i | 8.81974 | − | 5.04361i | − | 6.95342i | −0.0991579 | + | 7.99939i | 16.8064 | 1.33679 | + | 2.33764i | ||||||
35.7 | −1.35773 | − | 1.46853i | 0.832222 | −0.313139 | + | 3.98772i | − | 1.35598i | −1.12993 | − | 1.22214i | 13.3698i | 6.28124 | − | 4.95440i | −8.30741 | −1.99129 | + | 1.84105i | |||||||
35.8 | −1.35773 | + | 1.46853i | 0.832222 | −0.313139 | − | 3.98772i | 1.35598i | −1.12993 | + | 1.22214i | − | 13.3698i | 6.28124 | + | 4.95440i | −8.30741 | −1.99129 | − | 1.84105i | |||||||
35.9 | −1.03740 | − | 1.70991i | 0.111431 | −1.84759 | + | 3.54773i | − | 2.64262i | −0.115599 | − | 0.190537i | − | 5.52885i | 7.98300 | − | 0.521210i | −8.98758 | −4.51864 | + | 2.74146i | ||||||
35.10 | −1.03740 | + | 1.70991i | 0.111431 | −1.84759 | − | 3.54773i | 2.64262i | −0.115599 | + | 0.190537i | 5.52885i | 7.98300 | + | 0.521210i | −8.98758 | −4.51864 | − | 2.74146i | ||||||||
35.11 | −0.801563 | − | 1.83235i | 5.56462 | −2.71499 | + | 2.93748i | 6.17932i | −4.46039 | − | 10.1963i | 1.08834i | 7.55873 | + | 2.62023i | 21.9650 | 11.3227 | − | 4.95311i | ||||||||
35.12 | −0.801563 | + | 1.83235i | 5.56462 | −2.71499 | − | 2.93748i | − | 6.17932i | −4.46039 | + | 10.1963i | − | 1.08834i | 7.55873 | − | 2.62023i | 21.9650 | 11.3227 | + | 4.95311i | ||||||
35.13 | −0.697886 | − | 1.87429i | −3.94276 | −3.02591 | + | 2.61608i | 4.71646i | 2.75160 | + | 7.38987i | − | 8.04857i | 7.01502 | + | 3.84570i | 6.54538 | 8.84000 | − | 3.29155i | |||||||
35.14 | −0.697886 | + | 1.87429i | −3.94276 | −3.02591 | − | 2.61608i | − | 4.71646i | 2.75160 | − | 7.38987i | 8.04857i | 7.01502 | − | 3.84570i | 6.54538 | 8.84000 | + | 3.29155i | |||||||
35.15 | 0.0317538 | − | 1.99975i | −4.33534 | −3.99798 | − | 0.126999i | − | 9.04730i | −0.137663 | + | 8.66959i | 6.61433i | −0.380917 | + | 7.99093i | 9.79518 | −18.0923 | − | 0.287286i | |||||||
35.16 | 0.0317538 | + | 1.99975i | −4.33534 | −3.99798 | + | 0.126999i | 9.04730i | −0.137663 | − | 8.66959i | − | 6.61433i | −0.380917 | − | 7.99093i | 9.79518 | −18.0923 | + | 0.287286i | |||||||
35.17 | 0.336933 | − | 1.97141i | 3.07030 | −3.77295 | − | 1.32847i | − | 5.04316i | 1.03449 | − | 6.05284i | − | 2.68396i | −3.89020 | + | 6.99045i | 0.426766 | −9.94217 | − | 1.69921i | ||||||
35.18 | 0.336933 | + | 1.97141i | 3.07030 | −3.77295 | + | 1.32847i | 5.04316i | 1.03449 | + | 6.05284i | 2.68396i | −3.89020 | − | 6.99045i | 0.426766 | −9.94217 | + | 1.69921i | ||||||||
35.19 | 0.376144 | − | 1.96431i | −0.688578 | −3.71703 | − | 1.47773i | 7.39528i | −0.259005 | + | 1.35258i | 10.6434i | −4.30085 | + | 6.74557i | −8.52586 | 14.5266 | + | 2.78169i | ||||||||
35.20 | 0.376144 | + | 1.96431i | −0.688578 | −3.71703 | + | 1.47773i | − | 7.39528i | −0.259005 | − | 1.35258i | − | 10.6434i | −4.30085 | − | 6.74557i | −8.52586 | 14.5266 | − | 2.78169i | ||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.d | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 136.3.f.a | ✓ | 32 |
4.b | odd | 2 | 1 | 544.3.f.a | 32 | ||
8.b | even | 2 | 1 | 544.3.f.a | 32 | ||
8.d | odd | 2 | 1 | inner | 136.3.f.a | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
136.3.f.a | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
136.3.f.a | ✓ | 32 | 8.d | odd | 2 | 1 | inner |
544.3.f.a | 32 | 4.b | odd | 2 | 1 | ||
544.3.f.a | 32 | 8.b | even | 2 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(136, [\chi])\).