Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8028,2,Mod(4013,8028)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8028, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8028.4013");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8028 = 2^{2} \cdot 3^{2} \cdot 223 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8028.h (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: | \(64.1039027427\) |
Analytic rank: | \(0\) |
Dimension: | \(76\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4013.1 | 0 | 0 | 0 | −4.23105 | 0 | 2.43977 | 0 | 0 | 0 | ||||||||||||||||||
4013.2 | 0 | 0 | 0 | −4.23105 | 0 | 2.43977 | 0 | 0 | 0 | ||||||||||||||||||
4013.3 | 0 | 0 | 0 | −3.84036 | 0 | 1.75909 | 0 | 0 | 0 | ||||||||||||||||||
4013.4 | 0 | 0 | 0 | −3.84036 | 0 | 1.75909 | 0 | 0 | 0 | ||||||||||||||||||
4013.5 | 0 | 0 | 0 | −3.83180 | 0 | −1.12103 | 0 | 0 | 0 | ||||||||||||||||||
4013.6 | 0 | 0 | 0 | −3.83180 | 0 | −1.12103 | 0 | 0 | 0 | ||||||||||||||||||
4013.7 | 0 | 0 | 0 | −3.80256 | 0 | 0.208948 | 0 | 0 | 0 | ||||||||||||||||||
4013.8 | 0 | 0 | 0 | −3.80256 | 0 | 0.208948 | 0 | 0 | 0 | ||||||||||||||||||
4013.9 | 0 | 0 | 0 | −3.65965 | 0 | −4.62592 | 0 | 0 | 0 | ||||||||||||||||||
4013.10 | 0 | 0 | 0 | −3.65965 | 0 | −4.62592 | 0 | 0 | 0 | ||||||||||||||||||
4013.11 | 0 | 0 | 0 | −2.81288 | 0 | 3.64674 | 0 | 0 | 0 | ||||||||||||||||||
4013.12 | 0 | 0 | 0 | −2.81288 | 0 | 3.64674 | 0 | 0 | 0 | ||||||||||||||||||
4013.13 | 0 | 0 | 0 | −2.68681 | 0 | −3.11614 | 0 | 0 | 0 | ||||||||||||||||||
4013.14 | 0 | 0 | 0 | −2.68681 | 0 | −3.11614 | 0 | 0 | 0 | ||||||||||||||||||
4013.15 | 0 | 0 | 0 | −2.24499 | 0 | −1.57279 | 0 | 0 | 0 | ||||||||||||||||||
4013.16 | 0 | 0 | 0 | −2.24499 | 0 | −1.57279 | 0 | 0 | 0 | ||||||||||||||||||
4013.17 | 0 | 0 | 0 | −2.12158 | 0 | 2.81365 | 0 | 0 | 0 | ||||||||||||||||||
4013.18 | 0 | 0 | 0 | −2.12158 | 0 | 2.81365 | 0 | 0 | 0 | ||||||||||||||||||
4013.19 | 0 | 0 | 0 | −1.82157 | 0 | −3.59530 | 0 | 0 | 0 | ||||||||||||||||||
4013.20 | 0 | 0 | 0 | −1.82157 | 0 | −3.59530 | 0 | 0 | 0 | ||||||||||||||||||
See all 76 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
223.b | odd | 2 | 1 | inner |
669.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 8028.2.h.a | ✓ | 76 |
3.b | odd | 2 | 1 | inner | 8028.2.h.a | ✓ | 76 |
223.b | odd | 2 | 1 | inner | 8028.2.h.a | ✓ | 76 |
669.c | even | 2 | 1 | inner | 8028.2.h.a | ✓ | 76 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8028.2.h.a | ✓ | 76 | 1.a | even | 1 | 1 | trivial |
8028.2.h.a | ✓ | 76 | 3.b | odd | 2 | 1 | inner |
8028.2.h.a | ✓ | 76 | 223.b | odd | 2 | 1 | inner |
8028.2.h.a | ✓ | 76 | 669.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(8028, [\chi])\).