Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2000,1,Mod(399,2000)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2000, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 0, 1]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2000.399");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2000 = 2^{4} \cdot 5^{3} \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2000.x (of order \(10\), degree \(4\), 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: | \(0.998130025266\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{10})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 400) |
| Projective image: | \(D_{10}\) |
| Projective field: | Galois closure of 10.2.195312500000000.4 |
$q$-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 \(\left(\mathbb{Z}/2000\mathbb{Z}\right)^\times\).
| \(n\) | \(501\) | \(751\) | \(1377\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(\zeta_{10}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 399.1 |
|
0 | 0 | 0 | 0 | 0 | 0 | 0 | −0.309017 | − | 0.951057i | 0 | ||||||||||||||||||||||||||||
| 799.1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.809017 | − | 0.587785i | 0 | |||||||||||||||||||||||||||||
| 1199.1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.809017 | + | 0.587785i | 0 | |||||||||||||||||||||||||||||
| 1599.1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −0.309017 | + | 0.951057i | 0 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-1}) \) |
| 25.e | even | 10 | 1 | inner |
| 100.h | odd | 10 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2000.1.x.a | 4 | |
| 4.b | odd | 2 | 1 | CM | 2000.1.x.a | 4 | |
| 5.b | even | 2 | 1 | 400.1.x.a | ✓ | 4 | |
| 5.c | odd | 4 | 2 | 2000.1.z.a | 8 | ||
| 15.d | odd | 2 | 1 | 3600.1.ct.a | 4 | ||
| 20.d | odd | 2 | 1 | 400.1.x.a | ✓ | 4 | |
| 20.e | even | 4 | 2 | 2000.1.z.a | 8 | ||
| 25.d | even | 5 | 1 | 400.1.x.a | ✓ | 4 | |
| 25.e | even | 10 | 1 | inner | 2000.1.x.a | 4 | |
| 25.f | odd | 20 | 2 | 2000.1.z.a | 8 | ||
| 40.e | odd | 2 | 1 | 1600.1.bf.a | 4 | ||
| 40.f | even | 2 | 1 | 1600.1.bf.a | 4 | ||
| 60.h | even | 2 | 1 | 3600.1.ct.a | 4 | ||
| 75.j | odd | 10 | 1 | 3600.1.ct.a | 4 | ||
| 100.h | odd | 10 | 1 | inner | 2000.1.x.a | 4 | |
| 100.j | odd | 10 | 1 | 400.1.x.a | ✓ | 4 | |
| 100.l | even | 20 | 2 | 2000.1.z.a | 8 | ||
| 200.n | odd | 10 | 1 | 1600.1.bf.a | 4 | ||
| 200.t | even | 10 | 1 | 1600.1.bf.a | 4 | ||
| 300.n | even | 10 | 1 | 3600.1.ct.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 400.1.x.a | ✓ | 4 | 5.b | even | 2 | 1 | |
| 400.1.x.a | ✓ | 4 | 20.d | odd | 2 | 1 | |
| 400.1.x.a | ✓ | 4 | 25.d | even | 5 | 1 | |
| 400.1.x.a | ✓ | 4 | 100.j | odd | 10 | 1 | |
| 1600.1.bf.a | 4 | 40.e | odd | 2 | 1 | ||
| 1600.1.bf.a | 4 | 40.f | even | 2 | 1 | ||
| 1600.1.bf.a | 4 | 200.n | odd | 10 | 1 | ||
| 1600.1.bf.a | 4 | 200.t | even | 10 | 1 | ||
| 2000.1.x.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 2000.1.x.a | 4 | 4.b | odd | 2 | 1 | CM | |
| 2000.1.x.a | 4 | 25.e | even | 10 | 1 | inner | |
| 2000.1.x.a | 4 | 100.h | odd | 10 | 1 | inner | |
| 2000.1.z.a | 8 | 5.c | odd | 4 | 2 | ||
| 2000.1.z.a | 8 | 20.e | even | 4 | 2 | ||
| 2000.1.z.a | 8 | 25.f | odd | 20 | 2 | ||
| 2000.1.z.a | 8 | 100.l | even | 20 | 2 | ||
| 3600.1.ct.a | 4 | 15.d | odd | 2 | 1 | ||
| 3600.1.ct.a | 4 | 60.h | even | 2 | 1 | ||
| 3600.1.ct.a | 4 | 75.j | odd | 10 | 1 | ||
| 3600.1.ct.a | 4 | 300.n | even | 10 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2000, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} - 5T + 5 \)
|
| $17$ |
\( T^{4} - 5T + 5 \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( T^{4} \)
|
| $29$ |
\( T^{4} + 2 T^{3} + \cdots + 1 \)
|
| $31$ |
\( T^{4} \)
|
| $37$ |
\( T^{4} - 5 T^{3} + \cdots + 5 \)
|
| $41$ |
\( T^{4} - 2 T^{3} + \cdots + 1 \)
|
| $43$ |
\( T^{4} \)
|
| $47$ |
\( T^{4} \)
|
| $53$ |
\( T^{4} - 5 T^{3} + \cdots + 5 \)
|
| $59$ |
\( T^{4} \)
|
| $61$ |
\( T^{4} - 2 T^{3} + \cdots + 1 \)
|
| $67$ |
\( T^{4} \)
|
| $71$ |
\( T^{4} \)
|
| $73$ |
\( T^{4} - 5T + 5 \)
|
| $79$ |
\( T^{4} \)
|
| $83$ |
\( T^{4} \)
|
| $89$ |
\( T^{4} - 3 T^{3} + \cdots + 1 \)
|
| $97$ |
\( T^{4} - 5T + 5 \)
|
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