Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4004,2,Mod(2157,4004)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4004, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4004.2157");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4004.m (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: | \(31.9721009693\) |
Analytic rank: | \(0\) |
Dimension: | \(30\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2157.1 | 0 | −2.58184 | 0 | 1.48218i | 0 | 1.00000i | 0 | 3.66591 | 0 | ||||||||||||||||||
2157.2 | 0 | −2.58184 | 0 | − | 1.48218i | 0 | − | 1.00000i | 0 | 3.66591 | 0 | ||||||||||||||||
2157.3 | 0 | −2.50189 | 0 | − | 2.36927i | 0 | − | 1.00000i | 0 | 3.25946 | 0 | ||||||||||||||||
2157.4 | 0 | −2.50189 | 0 | 2.36927i | 0 | 1.00000i | 0 | 3.25946 | 0 | ||||||||||||||||||
2157.5 | 0 | −2.42937 | 0 | 1.20006i | 0 | 1.00000i | 0 | 2.90185 | 0 | ||||||||||||||||||
2157.6 | 0 | −2.42937 | 0 | − | 1.20006i | 0 | − | 1.00000i | 0 | 2.90185 | 0 | ||||||||||||||||
2157.7 | 0 | −2.41023 | 0 | − | 3.42759i | 0 | 1.00000i | 0 | 2.80920 | 0 | |||||||||||||||||
2157.8 | 0 | −2.41023 | 0 | 3.42759i | 0 | − | 1.00000i | 0 | 2.80920 | 0 | |||||||||||||||||
2157.9 | 0 | −1.49114 | 0 | − | 1.37882i | 0 | 1.00000i | 0 | −0.776492 | 0 | |||||||||||||||||
2157.10 | 0 | −1.49114 | 0 | 1.37882i | 0 | − | 1.00000i | 0 | −0.776492 | 0 | |||||||||||||||||
2157.11 | 0 | −0.822838 | 0 | − | 0.417533i | 0 | − | 1.00000i | 0 | −2.32294 | 0 | ||||||||||||||||
2157.12 | 0 | −0.822838 | 0 | 0.417533i | 0 | 1.00000i | 0 | −2.32294 | 0 | ||||||||||||||||||
2157.13 | 0 | −0.0797053 | 0 | 0.579061i | 0 | − | 1.00000i | 0 | −2.99365 | 0 | |||||||||||||||||
2157.14 | 0 | −0.0797053 | 0 | − | 0.579061i | 0 | 1.00000i | 0 | −2.99365 | 0 | |||||||||||||||||
2157.15 | 0 | 0.0840880 | 0 | − | 3.49341i | 0 | − | 1.00000i | 0 | −2.99293 | 0 | ||||||||||||||||
2157.16 | 0 | 0.0840880 | 0 | 3.49341i | 0 | 1.00000i | 0 | −2.99293 | 0 | ||||||||||||||||||
2157.17 | 0 | 0.705440 | 0 | − | 2.65594i | 0 | 1.00000i | 0 | −2.50236 | 0 | |||||||||||||||||
2157.18 | 0 | 0.705440 | 0 | 2.65594i | 0 | − | 1.00000i | 0 | −2.50236 | 0 | |||||||||||||||||
2157.19 | 0 | 0.726270 | 0 | − | 2.96350i | 0 | 1.00000i | 0 | −2.47253 | 0 | |||||||||||||||||
2157.20 | 0 | 0.726270 | 0 | 2.96350i | 0 | − | 1.00000i | 0 | −2.47253 | 0 | |||||||||||||||||
See all 30 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 4004.2.m.b | ✓ | 30 |
13.b | even | 2 | 1 | inner | 4004.2.m.b | ✓ | 30 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4004.2.m.b | ✓ | 30 | 1.a | even | 1 | 1 | trivial |
4004.2.m.b | ✓ | 30 | 13.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{15} - 27 T_{3}^{13} + 282 T_{3}^{11} - 18 T_{3}^{10} - 1437 T_{3}^{9} + 270 T_{3}^{8} + 3702 T_{3}^{7} + \cdots + 6 \) acting on \(S_{2}^{\mathrm{new}}(4004, [\chi])\).