Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6240,2,Mod(3121,6240)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6240, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6240.3121");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6240.w (of order \(2\), degree \(1\), not 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: | \(49.8266508613\) |
Analytic rank: | \(0\) |
Dimension: | \(26\) |
Twist minimal: | no (minimal twist has level 1560) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3121.1 | 0 | − | 1.00000i | 0 | − | 1.00000i | 0 | −4.98673 | 0 | −1.00000 | 0 | ||||||||||||||||
3121.2 | 0 | 1.00000i | 0 | 1.00000i | 0 | −4.98673 | 0 | −1.00000 | 0 | ||||||||||||||||||
3121.3 | 0 | − | 1.00000i | 0 | − | 1.00000i | 0 | −2.68192 | 0 | −1.00000 | 0 | ||||||||||||||||
3121.4 | 0 | 1.00000i | 0 | 1.00000i | 0 | −2.68192 | 0 | −1.00000 | 0 | ||||||||||||||||||
3121.5 | 0 | − | 1.00000i | 0 | − | 1.00000i | 0 | 0.648833 | 0 | −1.00000 | 0 | ||||||||||||||||
3121.6 | 0 | 1.00000i | 0 | 1.00000i | 0 | 0.648833 | 0 | −1.00000 | 0 | ||||||||||||||||||
3121.7 | 0 | − | 1.00000i | 0 | − | 1.00000i | 0 | −2.60702 | 0 | −1.00000 | 0 | ||||||||||||||||
3121.8 | 0 | 1.00000i | 0 | 1.00000i | 0 | −2.60702 | 0 | −1.00000 | 0 | ||||||||||||||||||
3121.9 | 0 | − | 1.00000i | 0 | − | 1.00000i | 0 | −0.892698 | 0 | −1.00000 | 0 | ||||||||||||||||
3121.10 | 0 | 1.00000i | 0 | 1.00000i | 0 | −0.892698 | 0 | −1.00000 | 0 | ||||||||||||||||||
3121.11 | 0 | − | 1.00000i | 0 | − | 1.00000i | 0 | 2.17617 | 0 | −1.00000 | 0 | ||||||||||||||||
3121.12 | 0 | 1.00000i | 0 | 1.00000i | 0 | 2.17617 | 0 | −1.00000 | 0 | ||||||||||||||||||
3121.13 | 0 | − | 1.00000i | 0 | − | 1.00000i | 0 | −1.96726 | 0 | −1.00000 | 0 | ||||||||||||||||
3121.14 | 0 | 1.00000i | 0 | 1.00000i | 0 | −1.96726 | 0 | −1.00000 | 0 | ||||||||||||||||||
3121.15 | 0 | − | 1.00000i | 0 | − | 1.00000i | 0 | 1.82026 | 0 | −1.00000 | 0 | ||||||||||||||||
3121.16 | 0 | 1.00000i | 0 | 1.00000i | 0 | 1.82026 | 0 | −1.00000 | 0 | ||||||||||||||||||
3121.17 | 0 | − | 1.00000i | 0 | − | 1.00000i | 0 | 2.13154 | 0 | −1.00000 | 0 | ||||||||||||||||
3121.18 | 0 | 1.00000i | 0 | 1.00000i | 0 | 2.13154 | 0 | −1.00000 | 0 | ||||||||||||||||||
3121.19 | 0 | − | 1.00000i | 0 | − | 1.00000i | 0 | 4.44152 | 0 | −1.00000 | 0 | ||||||||||||||||
3121.20 | 0 | 1.00000i | 0 | 1.00000i | 0 | 4.44152 | 0 | −1.00000 | 0 | ||||||||||||||||||
See all 26 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 6240.2.w.h | 26 | |
4.b | odd | 2 | 1 | 1560.2.w.h | ✓ | 26 | |
8.b | even | 2 | 1 | inner | 6240.2.w.h | 26 | |
8.d | odd | 2 | 1 | 1560.2.w.h | ✓ | 26 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1560.2.w.h | ✓ | 26 | 4.b | odd | 2 | 1 | |
1560.2.w.h | ✓ | 26 | 8.d | odd | 2 | 1 | |
6240.2.w.h | 26 | 1.a | even | 1 | 1 | trivial | |
6240.2.w.h | 26 | 8.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(6240, [\chi])\):
\( T_{7}^{13} - 2 T_{7}^{12} - 54 T_{7}^{11} + 108 T_{7}^{10} + 1053 T_{7}^{9} - 2006 T_{7}^{8} + \cdots - 57856 \) |
\( T_{11}^{26} + 152 T_{11}^{24} + 10290 T_{11}^{22} + 410156 T_{11}^{20} + 10718305 T_{11}^{18} + \cdots + 1379516022784 \) |