Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1340,3,Mod(669,1340)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1340, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1340.669");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1340 = 2^{2} \cdot 5 \cdot 67 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 1340.g (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: | \(36.5123554243\) |
Analytic rank: | \(0\) |
Dimension: | \(68\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
669.1 | 0 | −5.92367 | 0 | 2.53183 | + | 4.31160i | 0 | −4.10255 | 0 | 26.0899 | 0 | ||||||||||||||||
669.2 | 0 | −5.92367 | 0 | 2.53183 | − | 4.31160i | 0 | −4.10255 | 0 | 26.0899 | 0 | ||||||||||||||||
669.3 | 0 | −5.25359 | 0 | −4.88691 | + | 1.05742i | 0 | 4.79083 | 0 | 18.6003 | 0 | ||||||||||||||||
669.4 | 0 | −5.25359 | 0 | −4.88691 | − | 1.05742i | 0 | 4.79083 | 0 | 18.6003 | 0 | ||||||||||||||||
669.5 | 0 | −4.80687 | 0 | −0.0438558 | − | 4.99981i | 0 | 7.91446 | 0 | 14.1060 | 0 | ||||||||||||||||
669.6 | 0 | −4.80687 | 0 | −0.0438558 | + | 4.99981i | 0 | 7.91446 | 0 | 14.1060 | 0 | ||||||||||||||||
669.7 | 0 | −4.51228 | 0 | 3.78719 | − | 3.26453i | 0 | 13.1927 | 0 | 11.3607 | 0 | ||||||||||||||||
669.8 | 0 | −4.51228 | 0 | 3.78719 | + | 3.26453i | 0 | 13.1927 | 0 | 11.3607 | 0 | ||||||||||||||||
669.9 | 0 | −4.43361 | 0 | −4.17384 | − | 2.75300i | 0 | −7.13124 | 0 | 10.6569 | 0 | ||||||||||||||||
669.10 | 0 | −4.43361 | 0 | −4.17384 | + | 2.75300i | 0 | −7.13124 | 0 | 10.6569 | 0 | ||||||||||||||||
669.11 | 0 | −4.21016 | 0 | −2.49779 | + | 4.33140i | 0 | −11.5732 | 0 | 8.72545 | 0 | ||||||||||||||||
669.12 | 0 | −4.21016 | 0 | −2.49779 | − | 4.33140i | 0 | −11.5732 | 0 | 8.72545 | 0 | ||||||||||||||||
669.13 | 0 | −3.95913 | 0 | 4.75828 | + | 1.53581i | 0 | −6.32946 | 0 | 6.67473 | 0 | ||||||||||||||||
669.14 | 0 | −3.95913 | 0 | 4.75828 | − | 1.53581i | 0 | −6.32946 | 0 | 6.67473 | 0 | ||||||||||||||||
669.15 | 0 | −3.94848 | 0 | 4.78326 | + | 1.45617i | 0 | 0.855885 | 0 | 6.59049 | 0 | ||||||||||||||||
669.16 | 0 | −3.94848 | 0 | 4.78326 | − | 1.45617i | 0 | 0.855885 | 0 | 6.59049 | 0 | ||||||||||||||||
669.17 | 0 | −3.05585 | 0 | −3.31835 | + | 3.74012i | 0 | 8.59815 | 0 | 0.338244 | 0 | ||||||||||||||||
669.18 | 0 | −3.05585 | 0 | −3.31835 | − | 3.74012i | 0 | 8.59815 | 0 | 0.338244 | 0 | ||||||||||||||||
669.19 | 0 | −2.66092 | 0 | 0.602251 | + | 4.96360i | 0 | −3.70312 | 0 | −1.91951 | 0 | ||||||||||||||||
669.20 | 0 | −2.66092 | 0 | 0.602251 | − | 4.96360i | 0 | −3.70312 | 0 | −1.91951 | 0 | ||||||||||||||||
See all 68 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
67.b | odd | 2 | 1 | inner |
335.d | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1340.3.g.a | ✓ | 68 |
5.b | even | 2 | 1 | inner | 1340.3.g.a | ✓ | 68 |
67.b | odd | 2 | 1 | inner | 1340.3.g.a | ✓ | 68 |
335.d | odd | 2 | 1 | inner | 1340.3.g.a | ✓ | 68 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1340.3.g.a | ✓ | 68 | 1.a | even | 1 | 1 | trivial |
1340.3.g.a | ✓ | 68 | 5.b | even | 2 | 1 | inner |
1340.3.g.a | ✓ | 68 | 67.b | odd | 2 | 1 | inner |
1340.3.g.a | ✓ | 68 | 335.d | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(1340, [\chi])\).