Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4032,2,Mod(1583,4032)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4032, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 3, 2, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4032.1583");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4032.v (of order \(4\), degree \(2\), 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: | \(32.1956820950\) |
Analytic rank: | \(0\) |
Dimension: | \(36\) |
Relative dimension: | \(18\) over \(\Q(i)\) |
Twist minimal: | no (minimal twist has level 1008) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1583.1 | 0 | 0 | 0 | −2.98923 | − | 2.98923i | 0 | 1.00000 | 0 | 0 | 0 | ||||||||||||||||
1583.2 | 0 | 0 | 0 | −2.53823 | − | 2.53823i | 0 | 1.00000 | 0 | 0 | 0 | ||||||||||||||||
1583.3 | 0 | 0 | 0 | −2.28967 | − | 2.28967i | 0 | 1.00000 | 0 | 0 | 0 | ||||||||||||||||
1583.4 | 0 | 0 | 0 | −2.01396 | − | 2.01396i | 0 | 1.00000 | 0 | 0 | 0 | ||||||||||||||||
1583.5 | 0 | 0 | 0 | −1.68827 | − | 1.68827i | 0 | 1.00000 | 0 | 0 | 0 | ||||||||||||||||
1583.6 | 0 | 0 | 0 | −1.18126 | − | 1.18126i | 0 | 1.00000 | 0 | 0 | 0 | ||||||||||||||||
1583.7 | 0 | 0 | 0 | −0.871498 | − | 0.871498i | 0 | 1.00000 | 0 | 0 | 0 | ||||||||||||||||
1583.8 | 0 | 0 | 0 | −0.495166 | − | 0.495166i | 0 | 1.00000 | 0 | 0 | 0 | ||||||||||||||||
1583.9 | 0 | 0 | 0 | −0.270063 | − | 0.270063i | 0 | 1.00000 | 0 | 0 | 0 | ||||||||||||||||
1583.10 | 0 | 0 | 0 | 0.270063 | + | 0.270063i | 0 | 1.00000 | 0 | 0 | 0 | ||||||||||||||||
1583.11 | 0 | 0 | 0 | 0.495166 | + | 0.495166i | 0 | 1.00000 | 0 | 0 | 0 | ||||||||||||||||
1583.12 | 0 | 0 | 0 | 0.871498 | + | 0.871498i | 0 | 1.00000 | 0 | 0 | 0 | ||||||||||||||||
1583.13 | 0 | 0 | 0 | 1.18126 | + | 1.18126i | 0 | 1.00000 | 0 | 0 | 0 | ||||||||||||||||
1583.14 | 0 | 0 | 0 | 1.68827 | + | 1.68827i | 0 | 1.00000 | 0 | 0 | 0 | ||||||||||||||||
1583.15 | 0 | 0 | 0 | 2.01396 | + | 2.01396i | 0 | 1.00000 | 0 | 0 | 0 | ||||||||||||||||
1583.16 | 0 | 0 | 0 | 2.28967 | + | 2.28967i | 0 | 1.00000 | 0 | 0 | 0 | ||||||||||||||||
1583.17 | 0 | 0 | 0 | 2.53823 | + | 2.53823i | 0 | 1.00000 | 0 | 0 | 0 | ||||||||||||||||
1583.18 | 0 | 0 | 0 | 2.98923 | + | 2.98923i | 0 | 1.00000 | 0 | 0 | 0 | ||||||||||||||||
3599.1 | 0 | 0 | 0 | −2.98923 | + | 2.98923i | 0 | 1.00000 | 0 | 0 | 0 | ||||||||||||||||
3599.2 | 0 | 0 | 0 | −2.53823 | + | 2.53823i | 0 | 1.00000 | 0 | 0 | 0 | ||||||||||||||||
See all 36 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
16.f | odd | 4 | 1 | inner |
48.k | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 4032.2.v.d | 36 | |
3.b | odd | 2 | 1 | inner | 4032.2.v.d | 36 | |
4.b | odd | 2 | 1 | 1008.2.v.d | ✓ | 36 | |
12.b | even | 2 | 1 | 1008.2.v.d | ✓ | 36 | |
16.e | even | 4 | 1 | 1008.2.v.d | ✓ | 36 | |
16.f | odd | 4 | 1 | inner | 4032.2.v.d | 36 | |
48.i | odd | 4 | 1 | 1008.2.v.d | ✓ | 36 | |
48.k | even | 4 | 1 | inner | 4032.2.v.d | 36 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1008.2.v.d | ✓ | 36 | 4.b | odd | 2 | 1 | |
1008.2.v.d | ✓ | 36 | 12.b | even | 2 | 1 | |
1008.2.v.d | ✓ | 36 | 16.e | even | 4 | 1 | |
1008.2.v.d | ✓ | 36 | 48.i | odd | 4 | 1 | |
4032.2.v.d | 36 | 1.a | even | 1 | 1 | trivial | |
4032.2.v.d | 36 | 3.b | odd | 2 | 1 | inner | |
4032.2.v.d | 36 | 16.f | odd | 4 | 1 | inner | |
4032.2.v.d | 36 | 48.k | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4032, [\chi])\):
\( T_{5}^{36} + 704 T_{5}^{32} + 174256 T_{5}^{28} + 19305408 T_{5}^{24} + 985890144 T_{5}^{20} + \cdots + 1146228736 \) |
\( T_{11}^{36} + 2208 T_{11}^{32} + 1643312 T_{11}^{28} + 477101376 T_{11}^{24} + 50827259744 T_{11}^{20} + \cdots + 6423507767296 \) |