Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [55,4,Mod(32,55)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(55, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([1, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("55.32");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 55 = 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 55.e (of order \(4\), 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: | \(3.24510505032\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{11})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 5x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{4}]$ |
$q$-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
\( x^{4} - 5x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 2\nu ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 8\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{2} - 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 5 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{2} + 4\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/55\mathbb{Z}\right)^\times\).
| \(n\) | \(12\) | \(46\) |
| \(\chi(n)\) | \(-\beta_{1}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 32.1 |
|
0 | 0.683375 | − | 0.683375i | 8.00000i | −6.63325 | + | 9.00000i | 0 | 0 | 0 | 26.0660i | 0 | ||||||||||||||||||||||||||
| 32.2 | 0 | 7.31662 | − | 7.31662i | 8.00000i | 6.63325 | + | 9.00000i | 0 | 0 | 0 | − | 80.0660i | 0 | ||||||||||||||||||||||||||
| 43.1 | 0 | 0.683375 | + | 0.683375i | − | 8.00000i | −6.63325 | − | 9.00000i | 0 | 0 | 0 | − | 26.0660i | 0 | |||||||||||||||||||||||||
| 43.2 | 0 | 7.31662 | + | 7.31662i | − | 8.00000i | 6.63325 | − | 9.00000i | 0 | 0 | 0 | 80.0660i | 0 | ||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-11}) \) |
| 5.c | odd | 4 | 1 | inner |
| 55.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 55.4.e.a | ✓ | 4 |
| 5.c | odd | 4 | 1 | inner | 55.4.e.a | ✓ | 4 |
| 11.b | odd | 2 | 1 | CM | 55.4.e.a | ✓ | 4 |
| 55.e | even | 4 | 1 | inner | 55.4.e.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 55.4.e.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 55.4.e.a | ✓ | 4 | 5.c | odd | 4 | 1 | inner |
| 55.4.e.a | ✓ | 4 | 11.b | odd | 2 | 1 | CM |
| 55.4.e.a | ✓ | 4 | 55.e | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{4}^{\mathrm{new}}(55, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 16 T^{3} + \cdots + 100 \)
|
| $5$ |
\( T^{4} + 74T^{2} + 15625 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} - 1331)^{2} \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( T^{4} - 216 T^{3} + \cdots + 160528900 \)
|
| $29$ |
\( T^{4} \)
|
| $31$ |
\( (T^{2} - 3564)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 7577702500 \)
|
| $41$ |
\( T^{4} \)
|
| $43$ |
\( T^{4} \)
|
| $47$ |
\( T^{4} + \cdots + 42580322500 \)
|
| $53$ |
\( T^{4} + \cdots + 60954672100 \)
|
| $59$ |
\( (T^{2} + 303116)^{2} \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( T^{4} + \cdots + 183586540900 \)
|
| $71$ |
\( (T - 612)^{4} \)
|
| $73$ |
\( T^{4} \)
|
| $79$ |
\( T^{4} \)
|
| $83$ |
\( T^{4} \)
|
| $89$ |
\( (T^{2} + 2802276)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 3327669156100 \)
|
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