Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [690,2,Mod(689,690)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(690, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 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("690.689");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 690.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: | \(5.50967773947\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
689.1 | −1.00000 | −1.65921 | − | 0.497008i | 1.00000 | −1.15459 | + | 1.91492i | 1.65921 | + | 0.497008i | −3.15825 | −1.00000 | 2.50597 | + | 1.64928i | 1.15459 | − | 1.91492i | ||||||||
689.2 | −1.00000 | −1.65921 | − | 0.497008i | 1.00000 | 1.15459 | − | 1.91492i | 1.65921 | + | 0.497008i | 3.15825 | −1.00000 | 2.50597 | + | 1.64928i | −1.15459 | + | 1.91492i | ||||||||
689.3 | −1.00000 | −1.65921 | + | 0.497008i | 1.00000 | −1.15459 | − | 1.91492i | 1.65921 | − | 0.497008i | −3.15825 | −1.00000 | 2.50597 | − | 1.64928i | 1.15459 | + | 1.91492i | ||||||||
689.4 | −1.00000 | −1.65921 | + | 0.497008i | 1.00000 | 1.15459 | + | 1.91492i | 1.65921 | − | 0.497008i | 3.15825 | −1.00000 | 2.50597 | − | 1.64928i | −1.15459 | − | 1.91492i | ||||||||
689.5 | −1.00000 | −1.29400 | − | 1.15133i | 1.00000 | −2.22484 | − | 0.223804i | 1.29400 | + | 1.15133i | 0.666856 | −1.00000 | 0.348885 | + | 2.97964i | 2.22484 | + | 0.223804i | ||||||||
689.6 | −1.00000 | −1.29400 | − | 1.15133i | 1.00000 | 2.22484 | + | 0.223804i | 1.29400 | + | 1.15133i | −0.666856 | −1.00000 | 0.348885 | + | 2.97964i | −2.22484 | − | 0.223804i | ||||||||
689.7 | −1.00000 | −1.29400 | + | 1.15133i | 1.00000 | −2.22484 | + | 0.223804i | 1.29400 | − | 1.15133i | 0.666856 | −1.00000 | 0.348885 | − | 2.97964i | 2.22484 | − | 0.223804i | ||||||||
689.8 | −1.00000 | −1.29400 | + | 1.15133i | 1.00000 | 2.22484 | − | 0.223804i | 1.29400 | − | 1.15133i | −0.666856 | −1.00000 | 0.348885 | − | 2.97964i | −2.22484 | + | 0.223804i | ||||||||
689.9 | −1.00000 | −0.250553 | − | 1.71383i | 1.00000 | −0.545034 | − | 2.16863i | 0.250553 | + | 1.71383i | 1.86244 | −1.00000 | −2.87445 | + | 0.858813i | 0.545034 | + | 2.16863i | ||||||||
689.10 | −1.00000 | −0.250553 | − | 1.71383i | 1.00000 | 0.545034 | + | 2.16863i | 0.250553 | + | 1.71383i | −1.86244 | −1.00000 | −2.87445 | + | 0.858813i | −0.545034 | − | 2.16863i | ||||||||
689.11 | −1.00000 | −0.250553 | + | 1.71383i | 1.00000 | −0.545034 | + | 2.16863i | 0.250553 | − | 1.71383i | 1.86244 | −1.00000 | −2.87445 | − | 0.858813i | 0.545034 | − | 2.16863i | ||||||||
689.12 | −1.00000 | −0.250553 | + | 1.71383i | 1.00000 | 0.545034 | − | 2.16863i | 0.250553 | − | 1.71383i | −1.86244 | −1.00000 | −2.87445 | − | 0.858813i | −0.545034 | + | 2.16863i | ||||||||
689.13 | −1.00000 | 0.571841 | − | 1.63493i | 1.00000 | −2.12720 | + | 0.689226i | −0.571841 | + | 1.63493i | 1.28874 | −1.00000 | −2.34600 | − | 1.86984i | 2.12720 | − | 0.689226i | ||||||||
689.14 | −1.00000 | 0.571841 | − | 1.63493i | 1.00000 | 2.12720 | − | 0.689226i | −0.571841 | + | 1.63493i | −1.28874 | −1.00000 | −2.34600 | − | 1.86984i | −2.12720 | + | 0.689226i | ||||||||
689.15 | −1.00000 | 0.571841 | + | 1.63493i | 1.00000 | −2.12720 | − | 0.689226i | −0.571841 | − | 1.63493i | 1.28874 | −1.00000 | −2.34600 | + | 1.86984i | 2.12720 | + | 0.689226i | ||||||||
689.16 | −1.00000 | 0.571841 | + | 1.63493i | 1.00000 | 2.12720 | + | 0.689226i | −0.571841 | − | 1.63493i | −1.28874 | −1.00000 | −2.34600 | + | 1.86984i | −2.12720 | − | 0.689226i | ||||||||
689.17 | −1.00000 | 1.44698 | − | 0.951967i | 1.00000 | −1.42959 | − | 1.71938i | −1.44698 | + | 0.951967i | −4.73682 | −1.00000 | 1.18752 | − | 2.75496i | 1.42959 | + | 1.71938i | ||||||||
689.18 | −1.00000 | 1.44698 | − | 0.951967i | 1.00000 | 1.42959 | + | 1.71938i | −1.44698 | + | 0.951967i | 4.73682 | −1.00000 | 1.18752 | − | 2.75496i | −1.42959 | − | 1.71938i | ||||||||
689.19 | −1.00000 | 1.44698 | + | 0.951967i | 1.00000 | −1.42959 | + | 1.71938i | −1.44698 | − | 0.951967i | −4.73682 | −1.00000 | 1.18752 | + | 2.75496i | 1.42959 | − | 1.71938i | ||||||||
689.20 | −1.00000 | 1.44698 | + | 0.951967i | 1.00000 | 1.42959 | − | 1.71938i | −1.44698 | − | 0.951967i | 4.73682 | −1.00000 | 1.18752 | + | 2.75496i | −1.42959 | + | 1.71938i | ||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
15.d | odd | 2 | 1 | inner |
23.b | odd | 2 | 1 | inner |
345.h | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 690.2.h.a | ✓ | 24 |
3.b | odd | 2 | 1 | 690.2.h.b | yes | 24 | |
5.b | even | 2 | 1 | 690.2.h.b | yes | 24 | |
15.d | odd | 2 | 1 | inner | 690.2.h.a | ✓ | 24 |
23.b | odd | 2 | 1 | inner | 690.2.h.a | ✓ | 24 |
69.c | even | 2 | 1 | 690.2.h.b | yes | 24 | |
115.c | odd | 2 | 1 | 690.2.h.b | yes | 24 | |
345.h | even | 2 | 1 | inner | 690.2.h.a | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
690.2.h.a | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
690.2.h.a | ✓ | 24 | 15.d | odd | 2 | 1 | inner |
690.2.h.a | ✓ | 24 | 23.b | odd | 2 | 1 | inner |
690.2.h.a | ✓ | 24 | 345.h | even | 2 | 1 | inner |
690.2.h.b | yes | 24 | 3.b | odd | 2 | 1 | |
690.2.h.b | yes | 24 | 5.b | even | 2 | 1 | |
690.2.h.b | yes | 24 | 69.c | even | 2 | 1 | |
690.2.h.b | yes | 24 | 115.c | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{47}^{6} - 4T_{47}^{5} - 132T_{47}^{4} + 472T_{47}^{3} + 2368T_{47}^{2} - 12288T_{47} + 13824 \) acting on \(S_{2}^{\mathrm{new}}(690, [\chi])\).