Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1550,2,Mod(249,1550)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1550, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1550.249");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1550 = 2 \cdot 5^{2} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1550.b (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: | \(12.3768123133\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{6})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 310) |
| 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
\( x^{4} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 3\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 3\nu ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3\beta_{3} + 3\beta_{2} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1550\mathbb{Z}\right)^\times\).
| \(n\) | \(251\) | \(1427\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 249.1 |
|
− | 1.00000i | − | 2.44949i | −1.00000 | 0 | −2.44949 | − | 2.00000i | 1.00000i | −3.00000 | 0 | |||||||||||||||||||||||||||
| 249.2 | − | 1.00000i | 2.44949i | −1.00000 | 0 | 2.44949 | − | 2.00000i | 1.00000i | −3.00000 | 0 | |||||||||||||||||||||||||||||
| 249.3 | 1.00000i | − | 2.44949i | −1.00000 | 0 | 2.44949 | 2.00000i | − | 1.00000i | −3.00000 | 0 | |||||||||||||||||||||||||||||
| 249.4 | 1.00000i | 2.44949i | −1.00000 | 0 | −2.44949 | 2.00000i | − | 1.00000i | −3.00000 | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1550.2.b.g | 4 | |
| 5.b | even | 2 | 1 | inner | 1550.2.b.g | 4 | |
| 5.c | odd | 4 | 1 | 310.2.a.d | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 1550.2.a.i | 2 | ||
| 15.e | even | 4 | 1 | 2790.2.a.bg | 2 | ||
| 20.e | even | 4 | 1 | 2480.2.a.q | 2 | ||
| 40.i | odd | 4 | 1 | 9920.2.a.bo | 2 | ||
| 40.k | even | 4 | 1 | 9920.2.a.bq | 2 | ||
| 155.f | even | 4 | 1 | 9610.2.a.f | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 310.2.a.d | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 1550.2.a.i | 2 | 5.c | odd | 4 | 1 | ||
| 1550.2.b.g | 4 | 1.a | even | 1 | 1 | trivial | |
| 1550.2.b.g | 4 | 5.b | even | 2 | 1 | inner | |
| 2480.2.a.q | 2 | 20.e | even | 4 | 1 | ||
| 2790.2.a.bg | 2 | 15.e | even | 4 | 1 | ||
| 9610.2.a.f | 2 | 155.f | even | 4 | 1 | ||
| 9920.2.a.bo | 2 | 40.i | odd | 4 | 1 | ||
| 9920.2.a.bq | 2 | 40.k | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1550, [\chi])\):
|
\( T_{3}^{2} + 6 \)
|
|
\( T_{7}^{2} + 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{2} \)
|
| $3$ |
\( (T^{2} + 6)^{2} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} + 4)^{2} \)
|
| $11$ |
\( (T^{2} - 4 T - 2)^{2} \)
|
| $13$ |
\( T^{4} + 20T^{2} + 4 \)
|
| $17$ |
\( (T^{2} + 24)^{2} \)
|
| $19$ |
\( (T^{2} - 24)^{2} \)
|
| $23$ |
\( (T^{2} + 4)^{2} \)
|
| $29$ |
\( (T^{2} + 16 T + 58)^{2} \)
|
| $31$ |
\( (T + 1)^{4} \)
|
| $37$ |
\( T^{4} + 20T^{2} + 4 \)
|
| $41$ |
\( T^{4} \)
|
| $43$ |
\( T^{4} + 140T^{2} + 3364 \)
|
| $47$ |
\( (T^{2} + 36)^{2} \)
|
| $53$ |
\( T^{4} + 116T^{2} + 2500 \)
|
| $59$ |
\( (T^{2} + 8 T - 8)^{2} \)
|
| $61$ |
\( (T^{2} + 8 T + 10)^{2} \)
|
| $67$ |
\( T^{4} + 176T^{2} + 1600 \)
|
| $71$ |
\( (T^{2} - 96)^{2} \)
|
| $73$ |
\( (T^{2} + 16)^{2} \)
|
| $79$ |
\( (T^{2} - 24)^{2} \)
|
| $83$ |
\( T^{4} + 332 T^{2} + 17956 \)
|
| $89$ |
\( (T^{2} + 12 T - 60)^{2} \)
|
| $97$ |
\( T^{4} + 80T^{2} + 64 \)
|
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