Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [363,3,Mod(122,363)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(363, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("363.122");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 363 = 3 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 363.b (of order \(2\), degree \(1\), 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: | \(9.89103359628\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-7})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} + 9x^{2} - 8x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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
\( x^{4} - 2x^{3} + 9x^{2} - 8x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu^{2} - 2\nu + 8 \)
|
| \(\beta_{2}\) | \(=\) |
\( 4\nu^{3} - 6\nu^{2} + 32\nu - 15 \)
|
| \(\beta_{3}\) | \(=\) |
\( 4\nu^{3} - 6\nu^{2} + 34\nu - 16 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_{2} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - \beta_{2} + \beta _1 - 7 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -13\beta_{3} + 14\beta_{2} + 3\beta _1 - 22 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).
| \(n\) | \(122\) | \(244\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 122.1 |
|
− | 2.64575i | −1.00000 | − | 2.82843i | −3.00000 | − | 2.82843i | −7.48331 | + | 2.64575i | 7.48331 | − | 2.64575i | −7.00000 | + | 5.65685i | −7.48331 | |||||||||||||||||||||
| 122.2 | − | 2.64575i | −1.00000 | + | 2.82843i | −3.00000 | 2.82843i | 7.48331 | + | 2.64575i | −7.48331 | − | 2.64575i | −7.00000 | − | 5.65685i | 7.48331 | |||||||||||||||||||||||
| 122.3 | 2.64575i | −1.00000 | − | 2.82843i | −3.00000 | − | 2.82843i | 7.48331 | − | 2.64575i | −7.48331 | 2.64575i | −7.00000 | + | 5.65685i | 7.48331 | ||||||||||||||||||||||||
| 122.4 | 2.64575i | −1.00000 | + | 2.82843i | −3.00000 | 2.82843i | −7.48331 | − | 2.64575i | 7.48331 | 2.64575i | −7.00000 | − | 5.65685i | −7.48331 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 11.b | odd | 2 | 1 | inner |
| 33.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 363.3.b.i | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 363.3.b.i | ✓ | 4 |
| 11.b | odd | 2 | 1 | inner | 363.3.b.i | ✓ | 4 |
| 11.c | even | 5 | 4 | 363.3.h.k | 16 | ||
| 11.d | odd | 10 | 4 | 363.3.h.k | 16 | ||
| 33.d | even | 2 | 1 | inner | 363.3.b.i | ✓ | 4 |
| 33.f | even | 10 | 4 | 363.3.h.k | 16 | ||
| 33.h | odd | 10 | 4 | 363.3.h.k | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 363.3.b.i | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 363.3.b.i | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 363.3.b.i | ✓ | 4 | 11.b | odd | 2 | 1 | inner |
| 363.3.b.i | ✓ | 4 | 33.d | even | 2 | 1 | inner |
| 363.3.h.k | 16 | 11.c | even | 5 | 4 | ||
| 363.3.h.k | 16 | 11.d | odd | 10 | 4 | ||
| 363.3.h.k | 16 | 33.f | even | 10 | 4 | ||
| 363.3.h.k | 16 | 33.h | odd | 10 | 4 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(363, [\chi])\):
|
\( T_{2}^{2} + 7 \)
|
|
\( T_{5}^{2} + 8 \)
|
|
\( T_{7}^{2} - 56 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 7)^{2} \)
|
| $3$ |
\( (T^{2} + 2 T + 9)^{2} \)
|
| $5$ |
\( (T^{2} + 8)^{2} \)
|
| $7$ |
\( (T^{2} - 56)^{2} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( (T^{2} - 504)^{2} \)
|
| $17$ |
\( (T^{2} + 448)^{2} \)
|
| $19$ |
\( (T^{2} - 224)^{2} \)
|
| $23$ |
\( (T^{2} + 968)^{2} \)
|
| $29$ |
\( (T^{2} + 448)^{2} \)
|
| $31$ |
\( (T - 30)^{4} \)
|
| $37$ |
\( (T + 10)^{4} \)
|
| $41$ |
\( (T^{2} + 1792)^{2} \)
|
| $43$ |
\( (T^{2} - 224)^{2} \)
|
| $47$ |
\( (T^{2} + 1352)^{2} \)
|
| $53$ |
\( (T^{2} + 1800)^{2} \)
|
| $59$ |
\( (T^{2} + 1152)^{2} \)
|
| $61$ |
\( (T^{2} - 9464)^{2} \)
|
| $67$ |
\( (T + 42)^{4} \)
|
| $71$ |
\( (T^{2} + 4232)^{2} \)
|
| $73$ |
\( (T^{2} - 5600)^{2} \)
|
| $79$ |
\( (T^{2} - 504)^{2} \)
|
| $83$ |
\( (T^{2} + 448)^{2} \)
|
| $89$ |
\( (T^{2} + 3872)^{2} \)
|
| $97$ |
\( (T - 74)^{4} \)
|
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