Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2001,1,Mod(1172,2001)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2001, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 2, 1]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2001.1172");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2001 = 3 \cdot 23 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2001.i (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: | \(0.998629090279\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{12}\) |
| Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2001\mathbb{Z}\right)^\times\).
| \(n\) | \(553\) | \(668\) | \(1132\) |
| \(\chi(n)\) | \(\zeta_{12}^{3}\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1172.1 |
|
−1.36603 | + | 1.36603i | 0.866025 | − | 0.500000i | − | 2.73205i | 0 | −0.500000 | + | 1.86603i | 0 | 2.36603 | + | 2.36603i | 0.500000 | − | 0.866025i | 0 | |||||||||||||||||||
| 1172.2 | 0.366025 | − | 0.366025i | −0.866025 | − | 0.500000i | 0.732051i | 0 | −0.500000 | + | 0.133975i | 0 | 0.633975 | + | 0.633975i | 0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||
| 1931.1 | −1.36603 | − | 1.36603i | 0.866025 | + | 0.500000i | 2.73205i | 0 | −0.500000 | − | 1.86603i | 0 | 2.36603 | − | 2.36603i | 0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||
| 1931.2 | 0.366025 | + | 0.366025i | −0.866025 | + | 0.500000i | − | 0.732051i | 0 | −0.500000 | − | 0.133975i | 0 | 0.633975 | − | 0.633975i | 0.500000 | − | 0.866025i | 0 | ||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 23.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-23}) \) |
| 87.f | even | 4 | 1 | inner |
| 2001.i | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2001.1.i.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | 2001.1.i.d | yes | 4 | |
| 23.b | odd | 2 | 1 | CM | 2001.1.i.c | ✓ | 4 |
| 29.c | odd | 4 | 1 | 2001.1.i.d | yes | 4 | |
| 69.c | even | 2 | 1 | 2001.1.i.d | yes | 4 | |
| 87.f | even | 4 | 1 | inner | 2001.1.i.c | ✓ | 4 |
| 667.f | even | 4 | 1 | 2001.1.i.d | yes | 4 | |
| 2001.i | odd | 4 | 1 | inner | 2001.1.i.c | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2001.1.i.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 2001.1.i.c | ✓ | 4 | 23.b | odd | 2 | 1 | CM |
| 2001.1.i.c | ✓ | 4 | 87.f | even | 4 | 1 | inner |
| 2001.1.i.c | ✓ | 4 | 2001.i | odd | 4 | 1 | inner |
| 2001.1.i.d | yes | 4 | 3.b | odd | 2 | 1 | |
| 2001.1.i.d | yes | 4 | 29.c | odd | 4 | 1 | |
| 2001.1.i.d | yes | 4 | 69.c | even | 2 | 1 | |
| 2001.1.i.d | yes | 4 | 667.f | even | 4 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 2T_{2}^{3} + 2T_{2}^{2} - 2T_{2} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(2001, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 2 T^{3} + \cdots + 1 \)
|
| $3$ |
\( T^{4} - T^{2} + 1 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( (T^{2} + 1)^{2} \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( (T^{2} + 1)^{2} \)
|
| $29$ |
\( T^{4} - T^{2} + 1 \)
|
| $31$ |
\( T^{4} - 2 T^{3} + \cdots + 1 \)
|
| $37$ |
\( T^{4} \)
|
| $41$ |
\( T^{4} - 2 T^{3} + \cdots + 1 \)
|
| $43$ |
\( T^{4} \)
|
| $47$ |
\( T^{4} + 2 T^{3} + \cdots + 1 \)
|
| $53$ |
\( T^{4} \)
|
| $59$ |
\( (T^{2} + 4)^{2} \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( T^{4} \)
|
| $71$ |
\( (T^{2} - 3)^{2} \)
|
| $73$ |
\( T^{4} - 2 T^{3} + \cdots + 1 \)
|
| $79$ |
\( T^{4} \)
|
| $83$ |
\( T^{4} \)
|
| $89$ |
\( T^{4} \)
|
| $97$ |
\( T^{4} \)
|
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