Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [289,2,Mod(288,289)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(289, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("289.288");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 289 = 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 289.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: | \(2.30767661842\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.419904.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 6x^{4} + 9x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 6x^{4} + 9x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 3\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} + 4\nu^{2} + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} + 5\nu^{3} + 5\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} - 4\beta_{2} + 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} - 5\beta_{3} + 10\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/289\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 288.1 |
|
−1.53209 | − | 1.34730i | 0.347296 | − | 3.53209i | 2.06418i | − | 0.347296i | 2.53209 | 1.18479 | 5.41147i | |||||||||||||||||||||||||||||||||
| 288.2 | −1.53209 | 1.34730i | 0.347296 | 3.53209i | − | 2.06418i | 0.347296i | 2.53209 | 1.18479 | − | 5.41147i | |||||||||||||||||||||||||||||||||||
| 288.3 | −0.347296 | − | 0.879385i | −1.87939 | 2.34730i | 0.305407i | − | 1.87939i | 1.34730 | 2.22668 | − | 0.815207i | ||||||||||||||||||||||||||||||||||
| 288.4 | −0.347296 | 0.879385i | −1.87939 | − | 2.34730i | − | 0.305407i | 1.87939i | 1.34730 | 2.22668 | 0.815207i | |||||||||||||||||||||||||||||||||||
| 288.5 | 1.87939 | − | 2.53209i | 1.53209 | − | 0.120615i | − | 4.75877i | − | 1.53209i | −0.879385 | −3.41147 | − | 0.226682i | ||||||||||||||||||||||||||||||||
| 288.6 | 1.87939 | 2.53209i | 1.53209 | 0.120615i | 4.75877i | 1.53209i | −0.879385 | −3.41147 | 0.226682i | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 289.2.b.d | 6 | |
| 17.b | even | 2 | 1 | inner | 289.2.b.d | 6 | |
| 17.c | even | 4 | 1 | 289.2.a.d | ✓ | 3 | |
| 17.c | even | 4 | 1 | 289.2.a.e | yes | 3 | |
| 17.d | even | 8 | 4 | 289.2.c.d | 12 | ||
| 17.e | odd | 16 | 8 | 289.2.d.f | 24 | ||
| 51.f | odd | 4 | 1 | 2601.2.a.w | 3 | ||
| 51.f | odd | 4 | 1 | 2601.2.a.x | 3 | ||
| 68.f | odd | 4 | 1 | 4624.2.a.bd | 3 | ||
| 68.f | odd | 4 | 1 | 4624.2.a.bg | 3 | ||
| 85.j | even | 4 | 1 | 7225.2.a.s | 3 | ||
| 85.j | even | 4 | 1 | 7225.2.a.t | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 289.2.a.d | ✓ | 3 | 17.c | even | 4 | 1 | |
| 289.2.a.e | yes | 3 | 17.c | even | 4 | 1 | |
| 289.2.b.d | 6 | 1.a | even | 1 | 1 | trivial | |
| 289.2.b.d | 6 | 17.b | even | 2 | 1 | inner | |
| 289.2.c.d | 12 | 17.d | even | 8 | 4 | ||
| 289.2.d.f | 24 | 17.e | odd | 16 | 8 | ||
| 2601.2.a.w | 3 | 51.f | odd | 4 | 1 | ||
| 2601.2.a.x | 3 | 51.f | odd | 4 | 1 | ||
| 4624.2.a.bd | 3 | 68.f | odd | 4 | 1 | ||
| 4624.2.a.bg | 3 | 68.f | odd | 4 | 1 | ||
| 7225.2.a.s | 3 | 85.j | even | 4 | 1 | ||
| 7225.2.a.t | 3 | 85.j | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 3T_{2} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(289, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{3} - 3 T - 1)^{2} \)
|
| $3$ |
\( T^{6} + 9 T^{4} + \cdots + 9 \)
|
| $5$ |
\( T^{6} + 18 T^{4} + \cdots + 1 \)
|
| $7$ |
\( T^{6} + 6 T^{4} + \cdots + 1 \)
|
| $11$ |
\( T^{6} + 36 T^{4} + \cdots + 576 \)
|
| $13$ |
\( (T^{3} - 6 T^{2} - 9 T + 71)^{2} \)
|
| $17$ |
\( T^{6} \)
|
| $19$ |
\( (T^{3} - 3 T - 1)^{2} \)
|
| $23$ |
\( T^{6} + 66 T^{4} + \cdots + 1369 \)
|
| $29$ |
\( T^{6} + 18 T^{4} + \cdots + 81 \)
|
| $31$ |
\( T^{6} + 69 T^{4} + \cdots + 2809 \)
|
| $37$ |
\( T^{6} + 81 T^{4} + \cdots + 16129 \)
|
| $41$ |
\( T^{6} + 90 T^{4} + \cdots + 25281 \)
|
| $43$ |
\( (T^{3} + 15 T^{2} + \cdots - 89)^{2} \)
|
| $47$ |
\( (T^{3} + 21 T^{2} + \cdots + 321)^{2} \)
|
| $53$ |
\( (T^{3} - 18 T^{2} + \cdots + 72)^{2} \)
|
| $59$ |
\( (T^{3} - 9 T^{2} + \cdots + 171)^{2} \)
|
| $61$ |
\( T^{6} + 21 T^{4} + \cdots + 1 \)
|
| $67$ |
\( (T^{3} + 9 T^{2} + \cdots - 289)^{2} \)
|
| $71$ |
\( T^{6} + 225 T^{4} + \cdots + 361 \)
|
| $73$ |
\( T^{6} + 261 T^{4} + \cdots + 47961 \)
|
| $79$ |
\( T^{6} + 171 T^{4} + \cdots + 45369 \)
|
| $83$ |
\( (T^{3} + 9 T^{2} - 57 T + 71)^{2} \)
|
| $89$ |
\( (T^{3} + 15 T^{2} + \cdots - 613)^{2} \)
|
| $97$ |
\( T^{6} + 228 T^{4} + \cdots + 179776 \)
|
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