Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1248,2,Mod(31,1248)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1248, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 0, 0, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1248.31");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1248 = 2^{5} \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1248.bc (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: | \(9.96533017226\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
| \(\beta_{1}\) | \(=\) |
\( \zeta_{12}^{3} \)
|
| \(\beta_{2}\) | \(=\) |
\( -\zeta_{12}^{3} + 2\zeta_{12}^{2} + 2\zeta_{12} - 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\zeta_{12}^{3} - 2\zeta_{12}^{2} + 2\zeta_{12} + 1 \)
|
| \(\zeta_{12}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 2\beta_1 ) / 4 \)
|
| \(\zeta_{12}^{2}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} + 2 ) / 4 \)
|
| \(\zeta_{12}^{3}\) | \(=\) |
\( \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1248\mathbb{Z}\right)^\times\).
| \(n\) | \(703\) | \(769\) | \(833\) | \(1093\) |
| \(\chi(n)\) | \(-1\) | \(\beta_{1}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31.1 |
|
0 | − | 1.00000i | 0 | −0.732051 | + | 0.732051i | 0 | −1.00000 | + | 1.00000i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||
| 31.2 | 0 | − | 1.00000i | 0 | 2.73205 | − | 2.73205i | 0 | −1.00000 | + | 1.00000i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||
| 1087.1 | 0 | 1.00000i | 0 | −0.732051 | − | 0.732051i | 0 | −1.00000 | − | 1.00000i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||
| 1087.2 | 0 | 1.00000i | 0 | 2.73205 | + | 2.73205i | 0 | −1.00000 | − | 1.00000i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 52.f | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1248.2.bc.q | ✓ | 4 |
| 4.b | odd | 2 | 1 | 1248.2.bc.r | yes | 4 | |
| 13.d | odd | 4 | 1 | 1248.2.bc.r | yes | 4 | |
| 52.f | even | 4 | 1 | inner | 1248.2.bc.q | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1248.2.bc.q | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1248.2.bc.q | ✓ | 4 | 52.f | even | 4 | 1 | inner |
| 1248.2.bc.r | yes | 4 | 4.b | odd | 2 | 1 | |
| 1248.2.bc.r | yes | 4 | 13.d | odd | 4 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1248, [\chi])\):
|
\( T_{5}^{4} - 4T_{5}^{3} + 8T_{5}^{2} + 16T_{5} + 16 \)
|
|
\( T_{7}^{2} + 2T_{7} + 2 \)
|
|
\( T_{11}^{4} + 4T_{11}^{3} + 8T_{11}^{2} - 16T_{11} + 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} + 1)^{2} \)
|
| $5$ |
\( T^{4} - 4 T^{3} + \cdots + 16 \)
|
| $7$ |
\( (T^{2} + 2 T + 2)^{2} \)
|
| $11$ |
\( T^{4} + 4 T^{3} + \cdots + 16 \)
|
| $13$ |
\( T^{4} - 22T^{2} + 169 \)
|
| $17$ |
\( T^{4} + 56T^{2} + 16 \)
|
| $19$ |
\( (T^{2} + 6 T + 18)^{2} \)
|
| $23$ |
\( (T - 4)^{4} \)
|
| $29$ |
\( (T^{2} - 12)^{2} \)
|
| $31$ |
\( T^{4} + 12 T^{3} + \cdots + 36 \)
|
| $37$ |
\( T^{4} - 4 T^{3} + \cdots + 8836 \)
|
| $41$ |
\( T^{4} - 4 T^{3} + \cdots + 16 \)
|
| $43$ |
\( (T - 8)^{4} \)
|
| $47$ |
\( T^{4} - 12 T^{3} + \cdots + 144 \)
|
| $53$ |
\( (T^{2} - 8 T + 4)^{2} \)
|
| $59$ |
\( T^{4} - 4 T^{3} + \cdots + 21904 \)
|
| $61$ |
\( (T - 4)^{4} \)
|
| $67$ |
\( T^{4} - 12 T^{3} + \cdots + 36 \)
|
| $71$ |
\( T^{4} + 4 T^{3} + \cdots + 16 \)
|
| $73$ |
\( T^{4} - 4 T^{3} + \cdots + 484 \)
|
| $79$ |
\( (T^{2} + 16)^{2} \)
|
| $83$ |
\( T^{4} + 20 T^{3} + \cdots + 16 \)
|
| $89$ |
\( T^{4} - 12 T^{3} + \cdots + 17424 \)
|
| $97$ |
\( T^{4} - 12 T^{3} + \cdots + 36 \)
|
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