Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2100,2,Mod(1457,2100)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2100, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 2, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2100.1457");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 2100.s (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: | \(16.7685844245\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(i)\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1457.1 | 0 | −1.73156 | − | 0.0412661i | 0 | 0 | 0 | 0.707107 | + | 0.707107i | 0 | 2.99659 | + | 0.142909i | 0 | ||||||||||||
1457.2 | 0 | −1.69214 | + | 0.369667i | 0 | 0 | 0 | 0.707107 | + | 0.707107i | 0 | 2.72669 | − | 1.25106i | 0 | ||||||||||||
1457.3 | 0 | −1.63431 | + | 0.573626i | 0 | 0 | 0 | −0.707107 | − | 0.707107i | 0 | 2.34191 | − | 1.87496i | 0 | ||||||||||||
1457.4 | 0 | −1.28800 | − | 1.15804i | 0 | 0 | 0 | −0.707107 | − | 0.707107i | 0 | 0.317883 | + | 2.98311i | 0 | ||||||||||||
1457.5 | 0 | −1.15804 | − | 1.28800i | 0 | 0 | 0 | 0.707107 | + | 0.707107i | 0 | −0.317883 | + | 2.98311i | 0 | ||||||||||||
1457.6 | 0 | −0.573626 | + | 1.63431i | 0 | 0 | 0 | −0.707107 | − | 0.707107i | 0 | −2.34191 | − | 1.87496i | 0 | ||||||||||||
1457.7 | 0 | −0.369667 | + | 1.69214i | 0 | 0 | 0 | 0.707107 | + | 0.707107i | 0 | −2.72669 | − | 1.25106i | 0 | ||||||||||||
1457.8 | 0 | −0.0412661 | − | 1.73156i | 0 | 0 | 0 | −0.707107 | − | 0.707107i | 0 | −2.99659 | + | 0.142909i | 0 | ||||||||||||
1457.9 | 0 | 0.0412661 | + | 1.73156i | 0 | 0 | 0 | 0.707107 | + | 0.707107i | 0 | −2.99659 | + | 0.142909i | 0 | ||||||||||||
1457.10 | 0 | 0.369667 | − | 1.69214i | 0 | 0 | 0 | −0.707107 | − | 0.707107i | 0 | −2.72669 | − | 1.25106i | 0 | ||||||||||||
1457.11 | 0 | 0.573626 | − | 1.63431i | 0 | 0 | 0 | 0.707107 | + | 0.707107i | 0 | −2.34191 | − | 1.87496i | 0 | ||||||||||||
1457.12 | 0 | 1.15804 | + | 1.28800i | 0 | 0 | 0 | −0.707107 | − | 0.707107i | 0 | −0.317883 | + | 2.98311i | 0 | ||||||||||||
1457.13 | 0 | 1.28800 | + | 1.15804i | 0 | 0 | 0 | 0.707107 | + | 0.707107i | 0 | 0.317883 | + | 2.98311i | 0 | ||||||||||||
1457.14 | 0 | 1.63431 | − | 0.573626i | 0 | 0 | 0 | 0.707107 | + | 0.707107i | 0 | 2.34191 | − | 1.87496i | 0 | ||||||||||||
1457.15 | 0 | 1.69214 | − | 0.369667i | 0 | 0 | 0 | −0.707107 | − | 0.707107i | 0 | 2.72669 | − | 1.25106i | 0 | ||||||||||||
1457.16 | 0 | 1.73156 | + | 0.0412661i | 0 | 0 | 0 | −0.707107 | − | 0.707107i | 0 | 2.99659 | + | 0.142909i | 0 | ||||||||||||
1793.1 | 0 | −1.73156 | + | 0.0412661i | 0 | 0 | 0 | 0.707107 | − | 0.707107i | 0 | 2.99659 | − | 0.142909i | 0 | ||||||||||||
1793.2 | 0 | −1.69214 | − | 0.369667i | 0 | 0 | 0 | 0.707107 | − | 0.707107i | 0 | 2.72669 | + | 1.25106i | 0 | ||||||||||||
1793.3 | 0 | −1.63431 | − | 0.573626i | 0 | 0 | 0 | −0.707107 | + | 0.707107i | 0 | 2.34191 | + | 1.87496i | 0 | ||||||||||||
1793.4 | 0 | −1.28800 | + | 1.15804i | 0 | 0 | 0 | −0.707107 | + | 0.707107i | 0 | 0.317883 | − | 2.98311i | 0 | ||||||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
5.c | odd | 4 | 2 | inner |
15.d | odd | 2 | 1 | inner |
15.e | even | 4 | 2 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2100.2.s.c | ✓ | 32 |
3.b | odd | 2 | 1 | inner | 2100.2.s.c | ✓ | 32 |
5.b | even | 2 | 1 | inner | 2100.2.s.c | ✓ | 32 |
5.c | odd | 4 | 2 | inner | 2100.2.s.c | ✓ | 32 |
15.d | odd | 2 | 1 | inner | 2100.2.s.c | ✓ | 32 |
15.e | even | 4 | 2 | inner | 2100.2.s.c | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2100.2.s.c | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
2100.2.s.c | ✓ | 32 | 3.b | odd | 2 | 1 | inner |
2100.2.s.c | ✓ | 32 | 5.b | even | 2 | 1 | inner |
2100.2.s.c | ✓ | 32 | 5.c | odd | 4 | 2 | inner |
2100.2.s.c | ✓ | 32 | 15.d | odd | 2 | 1 | inner |
2100.2.s.c | ✓ | 32 | 15.e | even | 4 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{8} + 60T_{11}^{6} + 962T_{11}^{4} + 4452T_{11}^{2} + 6253 \) acting on \(S_{2}^{\mathrm{new}}(2100, [\chi])\).