Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3900,2,Mod(2257,3900)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3900, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 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("3900.2257");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 3900.bm (of order \(4\), degree \(2\), 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: | \(31.1416567883\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Relative dimension: | \(14\) over \(\Q(i)\) |
Twist minimal: | no (minimal twist has level 780) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2257.1 | 0 | −0.707107 | + | 0.707107i | 0 | 0 | 0 | −3.12196 | 0 | − | 1.00000i | 0 | |||||||||||||||
2257.2 | 0 | −0.707107 | + | 0.707107i | 0 | 0 | 0 | 2.80999 | 0 | − | 1.00000i | 0 | |||||||||||||||
2257.3 | 0 | −0.707107 | + | 0.707107i | 0 | 0 | 0 | 1.25764 | 0 | − | 1.00000i | 0 | |||||||||||||||
2257.4 | 0 | −0.707107 | + | 0.707107i | 0 | 0 | 0 | 1.66430 | 0 | − | 1.00000i | 0 | |||||||||||||||
2257.5 | 0 | −0.707107 | + | 0.707107i | 0 | 0 | 0 | 2.20183 | 0 | − | 1.00000i | 0 | |||||||||||||||
2257.6 | 0 | −0.707107 | + | 0.707107i | 0 | 0 | 0 | −2.82733 | 0 | − | 1.00000i | 0 | |||||||||||||||
2257.7 | 0 | −0.707107 | + | 0.707107i | 0 | 0 | 0 | −4.81289 | 0 | − | 1.00000i | 0 | |||||||||||||||
2257.8 | 0 | 0.707107 | − | 0.707107i | 0 | 0 | 0 | −4.69969 | 0 | − | 1.00000i | 0 | |||||||||||||||
2257.9 | 0 | 0.707107 | − | 0.707107i | 0 | 0 | 0 | 0.757795 | 0 | − | 1.00000i | 0 | |||||||||||||||
2257.10 | 0 | 0.707107 | − | 0.707107i | 0 | 0 | 0 | 0.657863 | 0 | − | 1.00000i | 0 | |||||||||||||||
2257.11 | 0 | 0.707107 | − | 0.707107i | 0 | 0 | 0 | 0.513301 | 0 | − | 1.00000i | 0 | |||||||||||||||
2257.12 | 0 | 0.707107 | − | 0.707107i | 0 | 0 | 0 | −2.13466 | 0 | − | 1.00000i | 0 | |||||||||||||||
2257.13 | 0 | 0.707107 | − | 0.707107i | 0 | 0 | 0 | 3.56717 | 0 | − | 1.00000i | 0 | |||||||||||||||
2257.14 | 0 | 0.707107 | − | 0.707107i | 0 | 0 | 0 | 4.16665 | 0 | − | 1.00000i | 0 | |||||||||||||||
2293.1 | 0 | −0.707107 | − | 0.707107i | 0 | 0 | 0 | −3.12196 | 0 | 1.00000i | 0 | ||||||||||||||||
2293.2 | 0 | −0.707107 | − | 0.707107i | 0 | 0 | 0 | 2.80999 | 0 | 1.00000i | 0 | ||||||||||||||||
2293.3 | 0 | −0.707107 | − | 0.707107i | 0 | 0 | 0 | 1.25764 | 0 | 1.00000i | 0 | ||||||||||||||||
2293.4 | 0 | −0.707107 | − | 0.707107i | 0 | 0 | 0 | 1.66430 | 0 | 1.00000i | 0 | ||||||||||||||||
2293.5 | 0 | −0.707107 | − | 0.707107i | 0 | 0 | 0 | 2.20183 | 0 | 1.00000i | 0 | ||||||||||||||||
2293.6 | 0 | −0.707107 | − | 0.707107i | 0 | 0 | 0 | −2.82733 | 0 | 1.00000i | 0 | ||||||||||||||||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
65.f | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3900.2.bm.b | 28 | |
5.b | even | 2 | 1 | 780.2.bm.a | yes | 28 | |
5.c | odd | 4 | 1 | 780.2.r.a | ✓ | 28 | |
5.c | odd | 4 | 1 | 3900.2.r.b | 28 | ||
13.d | odd | 4 | 1 | 3900.2.r.b | 28 | ||
15.d | odd | 2 | 1 | 2340.2.bp.i | 28 | ||
15.e | even | 4 | 1 | 2340.2.u.i | 28 | ||
65.f | even | 4 | 1 | inner | 3900.2.bm.b | 28 | |
65.g | odd | 4 | 1 | 780.2.r.a | ✓ | 28 | |
65.k | even | 4 | 1 | 780.2.bm.a | yes | 28 | |
195.j | odd | 4 | 1 | 2340.2.bp.i | 28 | ||
195.n | even | 4 | 1 | 2340.2.u.i | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
780.2.r.a | ✓ | 28 | 5.c | odd | 4 | 1 | |
780.2.r.a | ✓ | 28 | 65.g | odd | 4 | 1 | |
780.2.bm.a | yes | 28 | 5.b | even | 2 | 1 | |
780.2.bm.a | yes | 28 | 65.k | even | 4 | 1 | |
2340.2.u.i | 28 | 15.e | even | 4 | 1 | ||
2340.2.u.i | 28 | 195.n | even | 4 | 1 | ||
2340.2.bp.i | 28 | 15.d | odd | 2 | 1 | ||
2340.2.bp.i | 28 | 195.j | odd | 4 | 1 | ||
3900.2.r.b | 28 | 5.c | odd | 4 | 1 | ||
3900.2.r.b | 28 | 13.d | odd | 4 | 1 | ||
3900.2.bm.b | 28 | 1.a | even | 1 | 1 | trivial | |
3900.2.bm.b | 28 | 65.f | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{14} - 58 T_{7}^{12} + 40 T_{7}^{11} + 1241 T_{7}^{10} - 1656 T_{7}^{9} - 11584 T_{7}^{8} + \cdots - 20992 \) acting on \(S_{2}^{\mathrm{new}}(3900, [\chi])\).