Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [280,4,Mod(9,280)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(280, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 3, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("280.9");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 280 = 2^{3} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 280.bg (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: | \(16.5205348016\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| 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: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/280\mathbb{Z}\right)^\times\).
| \(n\) | \(57\) | \(71\) | \(141\) | \(241\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) | \(-\zeta_{12}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 9.1 |
|
0 | −0.866025 | + | 0.500000i | 0 | 6.16025 | + | 9.33013i | 0 | −0.866025 | + | 18.5000i | 0 | −13.0000 | + | 22.5167i | 0 | ||||||||||||||||||||||
| 9.2 | 0 | 0.866025 | − | 0.500000i | 0 | −11.1603 | − | 0.669873i | 0 | 0.866025 | − | 18.5000i | 0 | −13.0000 | + | 22.5167i | 0 | |||||||||||||||||||||||
| 249.1 | 0 | −0.866025 | − | 0.500000i | 0 | 6.16025 | − | 9.33013i | 0 | −0.866025 | − | 18.5000i | 0 | −13.0000 | − | 22.5167i | 0 | |||||||||||||||||||||||
| 249.2 | 0 | 0.866025 | + | 0.500000i | 0 | −11.1603 | + | 0.669873i | 0 | 0.866025 | + | 18.5000i | 0 | −13.0000 | − | 22.5167i | 0 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 7.c | even | 3 | 1 | inner |
| 35.j | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 280.4.bg.a | ✓ | 4 |
| 5.b | even | 2 | 1 | inner | 280.4.bg.a | ✓ | 4 |
| 7.c | even | 3 | 1 | inner | 280.4.bg.a | ✓ | 4 |
| 35.j | even | 6 | 1 | inner | 280.4.bg.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 280.4.bg.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 280.4.bg.a | ✓ | 4 | 5.b | even | 2 | 1 | inner |
| 280.4.bg.a | ✓ | 4 | 7.c | even | 3 | 1 | inner |
| 280.4.bg.a | ✓ | 4 | 35.j | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - T_{3}^{2} + 1 \)
acting on \(S_{4}^{\mathrm{new}}(280, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - T^{2} + 1 \)
|
| $5$ |
\( T^{4} + 10 T^{3} + \cdots + 15625 \)
|
| $7$ |
\( T^{4} + 683 T^{2} + 117649 \)
|
| $11$ |
\( (T^{2} - 16 T + 256)^{2} \)
|
| $13$ |
\( (T^{2} + 5476)^{2} \)
|
| $17$ |
\( T^{4} - 15876 T^{2} + 252047376 \)
|
| $19$ |
\( (T^{2} - 90 T + 8100)^{2} \)
|
| $23$ |
\( T^{4} - 6561 T^{2} + 43046721 \)
|
| $29$ |
\( (T + 51)^{4} \)
|
| $31$ |
\( (T^{2} - 266 T + 70756)^{2} \)
|
| $37$ |
\( T^{4} - 14400 T^{2} + 207360000 \)
|
| $41$ |
\( (T + 75)^{4} \)
|
| $43$ |
\( (T^{2} + 49729)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 27710263296 \)
|
| $53$ |
\( T^{4} + \cdots + 2428912656 \)
|
| $59$ |
\( (T^{2} - 582 T + 338724)^{2} \)
|
| $61$ |
\( (T^{2} + 603 T + 363609)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 29093783761 \)
|
| $71$ |
\( (T - 666)^{4} \)
|
| $73$ |
\( T^{4} - 5476 T^{2} + 29986576 \)
|
| $79$ |
\( (T^{2} + 374 T + 139876)^{2} \)
|
| $83$ |
\( (T^{2} + 149769)^{2} \)
|
| $89$ |
\( (T^{2} - 369 T + 136161)^{2} \)
|
| $97$ |
\( (T^{2} + 16384)^{2} \)
|
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