Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [456,2,Mod(11,456)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(456, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 3, 3, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("456.11");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 456 = 2^{3} \cdot 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 456.u (of order \(6\), 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: | \(3.64117833217\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-3})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{6}]$ |
$q$-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
\( x^{4} - 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/456\mathbb{Z}\right)^\times\).
| \(n\) | \(97\) | \(229\) | \(305\) | \(343\) |
| \(\chi(n)\) | \(-\beta_{2}\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 11.1 |
|
−1.22474 | − | 0.707107i | −1.00000 | − | 1.41421i | 1.00000 | + | 1.73205i | 0 | 0.224745 | + | 2.43916i | 0 | − | 2.82843i | −1.00000 | + | 2.82843i | 0 | |||||||||||||||||||
| 11.2 | 1.22474 | + | 0.707107i | −1.00000 | + | 1.41421i | 1.00000 | + | 1.73205i | 0 | −2.22474 | + | 1.02494i | 0 | 2.82843i | −1.00000 | − | 2.82843i | 0 | |||||||||||||||||||||
| 83.1 | −1.22474 | + | 0.707107i | −1.00000 | + | 1.41421i | 1.00000 | − | 1.73205i | 0 | 0.224745 | − | 2.43916i | 0 | 2.82843i | −1.00000 | − | 2.82843i | 0 | |||||||||||||||||||||
| 83.2 | 1.22474 | − | 0.707107i | −1.00000 | − | 1.41421i | 1.00000 | − | 1.73205i | 0 | −2.22474 | − | 1.02494i | 0 | − | 2.82843i | −1.00000 | + | 2.82843i | 0 | ||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-2}) \) |
| 57.h | odd | 6 | 1 | inner |
| 456.u | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 456.2.u.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 456.2.u.b | yes | 4 | |
| 8.d | odd | 2 | 1 | CM | 456.2.u.a | ✓ | 4 |
| 19.c | even | 3 | 1 | 456.2.u.b | yes | 4 | |
| 24.f | even | 2 | 1 | 456.2.u.b | yes | 4 | |
| 57.h | odd | 6 | 1 | inner | 456.2.u.a | ✓ | 4 |
| 152.k | odd | 6 | 1 | 456.2.u.b | yes | 4 | |
| 456.u | even | 6 | 1 | inner | 456.2.u.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 456.2.u.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 456.2.u.a | ✓ | 4 | 8.d | odd | 2 | 1 | CM |
| 456.2.u.a | ✓ | 4 | 57.h | odd | 6 | 1 | inner |
| 456.2.u.a | ✓ | 4 | 456.u | even | 6 | 1 | inner |
| 456.2.u.b | yes | 4 | 3.b | odd | 2 | 1 | |
| 456.2.u.b | yes | 4 | 19.c | even | 3 | 1 | |
| 456.2.u.b | yes | 4 | 24.f | even | 2 | 1 | |
| 456.2.u.b | yes | 4 | 152.k | odd | 6 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(456, [\chi])\):
|
\( T_{5} \)
|
|
\( T_{41}^{4} - 18T_{41}^{3} + 103T_{41}^{2} + 90T_{41} + 25 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 2T^{2} + 4 \)
|
| $3$ |
\( (T^{2} + 2 T + 3)^{2} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + 58T^{2} + 625 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} - 32T^{2} + 1024 \)
|
| $19$ |
\( T^{4} + 2 T^{3} + \cdots + 361 \)
|
| $23$ |
\( T^{4} \)
|
| $29$ |
\( T^{4} \)
|
| $31$ |
\( T^{4} \)
|
| $37$ |
\( T^{4} \)
|
| $41$ |
\( T^{4} - 18 T^{3} + \cdots + 25 \)
|
| $43$ |
\( (T^{2} - 10 T + 100)^{2} \)
|
| $47$ |
\( T^{4} \)
|
| $53$ |
\( T^{4} \)
|
| $59$ |
\( T^{4} - 18 T^{3} + \cdots + 529 \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( T^{4} - 14 T^{3} + \cdots + 25 \)
|
| $71$ |
\( T^{4} \)
|
| $73$ |
\( T^{4} - 2 T^{3} + \cdots + 46225 \)
|
| $79$ |
\( T^{4} \)
|
| $83$ |
\( T^{4} + 490 T^{2} + 58081 \)
|
| $89$ |
\( T^{4} - 32T^{2} + 1024 \)
|
| $97$ |
\( T^{4} + 10 T^{3} + \cdots + 36481 \)
|
| show more | |
| show less |