Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [450,3,Mod(449,450)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(450, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("450.449");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 450.b (of order \(2\), degree \(1\), not 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: | \(12.2616118962\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{8})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{29}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 18) |
| 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
| \(\beta_{1}\) | \(=\) |
\( 2\zeta_{8}^{2} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{8}^{3} + \zeta_{8} \)
|
| \(\beta_{3}\) | \(=\) |
\( -\zeta_{8}^{3} + \zeta_{8} \)
|
| \(\zeta_{8}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 2 \)
|
| \(\zeta_{8}^{2}\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\zeta_{8}^{3}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(127\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 449.1 |
|
−1.41421 | 0 | 2.00000 | 0 | 0 | − | 4.00000i | −2.82843 | 0 | 0 | |||||||||||||||||||||||||||||
| 449.2 | −1.41421 | 0 | 2.00000 | 0 | 0 | 4.00000i | −2.82843 | 0 | 0 | |||||||||||||||||||||||||||||||
| 449.3 | 1.41421 | 0 | 2.00000 | 0 | 0 | − | 4.00000i | 2.82843 | 0 | 0 | ||||||||||||||||||||||||||||||
| 449.4 | 1.41421 | 0 | 2.00000 | 0 | 0 | 4.00000i | 2.82843 | 0 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 5.b | even | 2 | 1 | inner |
| 15.d | odd | 2 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{2} + 16 \)
acting on \(S_{3}^{\mathrm{new}}(450, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} + 16)^{2} \)
|
| $11$ |
\( (T^{2} + 288)^{2} \)
|
| $13$ |
\( (T^{2} + 64)^{2} \)
|
| $17$ |
\( (T^{2} - 162)^{2} \)
|
| $19$ |
\( (T - 16)^{4} \)
|
| $23$ |
\( (T^{2} - 288)^{2} \)
|
| $29$ |
\( (T^{2} + 18)^{2} \)
|
| $31$ |
\( (T - 44)^{4} \)
|
| $37$ |
\( (T^{2} + 1156)^{2} \)
|
| $41$ |
\( (T^{2} + 2178)^{2} \)
|
| $43$ |
\( (T^{2} + 1600)^{2} \)
|
| $47$ |
\( (T^{2} - 7200)^{2} \)
|
| $53$ |
\( (T^{2} - 1458)^{2} \)
|
| $59$ |
\( (T^{2} + 1152)^{2} \)
|
| $61$ |
\( (T - 50)^{4} \)
|
| $67$ |
\( (T^{2} + 64)^{2} \)
|
| $71$ |
\( (T^{2} + 2592)^{2} \)
|
| $73$ |
\( (T^{2} + 256)^{2} \)
|
| $79$ |
\( (T - 76)^{4} \)
|
| $83$ |
\( (T^{2} - 14112)^{2} \)
|
| $89$ |
\( (T^{2} + 162)^{2} \)
|
| $97$ |
\( (T^{2} + 30976)^{2} \)
|
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