Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1000,2,Mod(749,1000)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1000, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1000.749");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1000 = 2^{3} \cdot 5^{3} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1000.f (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: | \(7.98504020213\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{5})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 3x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} + 3x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 2\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 2\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1000\mathbb{Z}\right)^\times\).
| \(n\) | \(377\) | \(501\) | \(751\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 749.1 |
|
−1.00000 | − | 1.00000i | −3.23607 | 2.00000i | 0 | 3.23607 | + | 3.23607i | 3.85410i | 2.00000 | − | 2.00000i | 7.47214 | 0 | ||||||||||||||||||||||||
| 749.2 | −1.00000 | − | 1.00000i | 1.23607 | 2.00000i | 0 | −1.23607 | − | 1.23607i | − | 2.85410i | 2.00000 | − | 2.00000i | −1.47214 | 0 | ||||||||||||||||||||||||
| 749.3 | −1.00000 | + | 1.00000i | −3.23607 | − | 2.00000i | 0 | 3.23607 | − | 3.23607i | − | 3.85410i | 2.00000 | + | 2.00000i | 7.47214 | 0 | |||||||||||||||||||||||
| 749.4 | −1.00000 | + | 1.00000i | 1.23607 | − | 2.00000i | 0 | −1.23607 | + | 1.23607i | 2.85410i | 2.00000 | + | 2.00000i | −1.47214 | 0 | ||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 40.f | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1000.2.f.a | 4 | |
| 4.b | odd | 2 | 1 | 4000.2.f.b | 4 | ||
| 5.b | even | 2 | 1 | 1000.2.f.b | 4 | ||
| 5.c | odd | 4 | 1 | 1000.2.d.a | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 1000.2.d.b | yes | 4 | |
| 8.b | even | 2 | 1 | 1000.2.f.b | 4 | ||
| 8.d | odd | 2 | 1 | 4000.2.f.a | 4 | ||
| 20.d | odd | 2 | 1 | 4000.2.f.a | 4 | ||
| 20.e | even | 4 | 1 | 4000.2.d.a | 4 | ||
| 20.e | even | 4 | 1 | 4000.2.d.b | 4 | ||
| 40.e | odd | 2 | 1 | 4000.2.f.b | 4 | ||
| 40.f | even | 2 | 1 | inner | 1000.2.f.a | 4 | |
| 40.i | odd | 4 | 1 | 1000.2.d.a | ✓ | 4 | |
| 40.i | odd | 4 | 1 | 1000.2.d.b | yes | 4 | |
| 40.k | even | 4 | 1 | 4000.2.d.a | 4 | ||
| 40.k | even | 4 | 1 | 4000.2.d.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1000.2.d.a | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 1000.2.d.a | ✓ | 4 | 40.i | odd | 4 | 1 | |
| 1000.2.d.b | yes | 4 | 5.c | odd | 4 | 1 | |
| 1000.2.d.b | yes | 4 | 40.i | odd | 4 | 1 | |
| 1000.2.f.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 1000.2.f.a | 4 | 40.f | even | 2 | 1 | inner | |
| 1000.2.f.b | 4 | 5.b | even | 2 | 1 | ||
| 1000.2.f.b | 4 | 8.b | even | 2 | 1 | ||
| 4000.2.d.a | 4 | 20.e | even | 4 | 1 | ||
| 4000.2.d.a | 4 | 40.k | even | 4 | 1 | ||
| 4000.2.d.b | 4 | 20.e | even | 4 | 1 | ||
| 4000.2.d.b | 4 | 40.k | even | 4 | 1 | ||
| 4000.2.f.a | 4 | 8.d | odd | 2 | 1 | ||
| 4000.2.f.a | 4 | 20.d | odd | 2 | 1 | ||
| 4000.2.f.b | 4 | 4.b | odd | 2 | 1 | ||
| 4000.2.f.b | 4 | 40.e | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 2T_{3} - 4 \)
acting on \(S_{2}^{\mathrm{new}}(1000, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T + 2)^{2} \)
|
| $3$ |
\( (T^{2} + 2 T - 4)^{2} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 23T^{2} + 121 \)
|
| $11$ |
\( T^{4} + 15T^{2} + 25 \)
|
| $13$ |
\( (T^{2} - 3 T + 1)^{2} \)
|
| $17$ |
\( (T^{2} + 4)^{2} \)
|
| $19$ |
\( T^{4} + 67T^{2} + 841 \)
|
| $23$ |
\( T^{4} + 27T^{2} + 81 \)
|
| $29$ |
\( T^{4} + 12T^{2} + 16 \)
|
| $31$ |
\( (T^{2} + 6 T + 4)^{2} \)
|
| $37$ |
\( (T^{2} - 11 T + 29)^{2} \)
|
| $41$ |
\( (T^{2} + 11 T + 19)^{2} \)
|
| $43$ |
\( (T^{2} + 2 T - 4)^{2} \)
|
| $47$ |
\( T^{4} + 163T^{2} + 1681 \)
|
| $53$ |
\( (T^{2} - 13 T + 41)^{2} \)
|
| $59$ |
\( T^{4} + 147T^{2} + 2401 \)
|
| $61$ |
\( T^{4} + 60T^{2} + 400 \)
|
| $67$ |
\( (T - 8)^{4} \)
|
| $71$ |
\( (T^{2} + 16 T + 44)^{2} \)
|
| $73$ |
\( T^{4} + 192T^{2} + 4096 \)
|
| $79$ |
\( (T^{2} - 30 T + 220)^{2} \)
|
| $83$ |
\( (T^{2} + 2 T - 44)^{2} \)
|
| $89$ |
\( (T^{2} + 15 T - 45)^{2} \)
|
| $97$ |
\( T^{4} + 368 T^{2} + 30976 \)
|
| show more | |
| show less |