Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [637,2,Mod(459,637)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(637, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("637.459");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 637 = 7^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 637.k (of order \(6\), degree \(2\), 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: | \(5.08647060876\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-7})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - x^{2} - 2x + 4 \)
|
| 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 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
\( x^{4} - x^{3} - x^{2} - 2x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + \nu^{2} - \nu - 2 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + \nu^{2} + \nu + 2 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + \beta _1 + 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/637\mathbb{Z}\right)^\times\).
| \(n\) | \(197\) | \(248\) |
| \(\chi(n)\) | \(\beta_{2}\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 459.1 |
|
− | 2.18890i | 0.895644 | + | 1.55130i | −2.79129 | −1.89564 | + | 1.09445i | 3.39564 | − | 1.96048i | 0 | 1.73205i | −0.104356 | + | 0.180750i | 2.39564 | + | 4.14938i | |||||||||||||||||||
| 459.2 | 0.456850i | −1.39564 | − | 2.41733i | 1.79129 | 0.395644 | − | 0.228425i | 1.10436 | − | 0.637600i | 0 | 1.73205i | −2.39564 | + | 4.14938i | 0.104356 | + | 0.180750i | |||||||||||||||||||||
| 569.1 | − | 0.456850i | −1.39564 | + | 2.41733i | 1.79129 | 0.395644 | + | 0.228425i | 1.10436 | + | 0.637600i | 0 | − | 1.73205i | −2.39564 | − | 4.14938i | 0.104356 | − | 0.180750i | |||||||||||||||||||
| 569.2 | 2.18890i | 0.895644 | − | 1.55130i | −2.79129 | −1.89564 | − | 1.09445i | 3.39564 | + | 1.96048i | 0 | − | 1.73205i | −0.104356 | − | 0.180750i | 2.39564 | − | 4.14938i | ||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 91.k | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 637.2.k.d | 4 | |
| 7.b | odd | 2 | 1 | 637.2.k.f | 4 | ||
| 7.c | even | 3 | 1 | 637.2.q.e | ✓ | 4 | |
| 7.c | even | 3 | 1 | 637.2.u.e | 4 | ||
| 7.d | odd | 6 | 1 | 637.2.q.f | yes | 4 | |
| 7.d | odd | 6 | 1 | 637.2.u.d | 4 | ||
| 13.e | even | 6 | 1 | 637.2.u.e | 4 | ||
| 91.k | even | 6 | 1 | inner | 637.2.k.d | 4 | |
| 91.l | odd | 6 | 1 | 637.2.k.f | 4 | ||
| 91.p | odd | 6 | 1 | 637.2.q.f | yes | 4 | |
| 91.t | odd | 6 | 1 | 637.2.u.d | 4 | ||
| 91.u | even | 6 | 1 | 637.2.q.e | ✓ | 4 | |
| 91.x | odd | 12 | 2 | 8281.2.a.bs | 4 | ||
| 91.ba | even | 12 | 2 | 8281.2.a.bq | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 637.2.k.d | 4 | 1.a | even | 1 | 1 | trivial | |
| 637.2.k.d | 4 | 91.k | even | 6 | 1 | inner | |
| 637.2.k.f | 4 | 7.b | odd | 2 | 1 | ||
| 637.2.k.f | 4 | 91.l | odd | 6 | 1 | ||
| 637.2.q.e | ✓ | 4 | 7.c | even | 3 | 1 | |
| 637.2.q.e | ✓ | 4 | 91.u | even | 6 | 1 | |
| 637.2.q.f | yes | 4 | 7.d | odd | 6 | 1 | |
| 637.2.q.f | yes | 4 | 91.p | odd | 6 | 1 | |
| 637.2.u.d | 4 | 7.d | odd | 6 | 1 | ||
| 637.2.u.d | 4 | 91.t | odd | 6 | 1 | ||
| 637.2.u.e | 4 | 7.c | even | 3 | 1 | ||
| 637.2.u.e | 4 | 13.e | even | 6 | 1 | ||
| 8281.2.a.bq | 4 | 91.ba | even | 12 | 2 | ||
| 8281.2.a.bs | 4 | 91.x | odd | 12 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\):
|
\( T_{2}^{4} + 5T_{2}^{2} + 1 \)
|
|
\( T_{3}^{4} + T_{3}^{3} + 6T_{3}^{2} - 5T_{3} + 25 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 5T^{2} + 1 \)
|
| $3$ |
\( T^{4} + T^{3} + \cdots + 25 \)
|
| $5$ |
\( T^{4} + 3 T^{3} + \cdots + 1 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} - 9 T^{3} + \cdots + 25 \)
|
| $13$ |
\( (T^{2} - 7 T + 13)^{2} \)
|
| $17$ |
\( (T - 3)^{4} \)
|
| $19$ |
\( T^{4} - 9 T^{3} + \cdots + 81 \)
|
| $23$ |
\( (T^{2} - 6 T - 12)^{2} \)
|
| $29$ |
\( T^{4} - 9 T^{3} + \cdots + 225 \)
|
| $31$ |
\( (T^{2} + 15 T + 75)^{2} \)
|
| $37$ |
\( (T^{2} + 48)^{2} \)
|
| $41$ |
\( T^{4} - 18 T^{3} + \cdots + 400 \)
|
| $43$ |
\( T^{4} + 5 T^{3} + \cdots + 1681 \)
|
| $47$ |
\( T^{4} + 24 T^{3} + \cdots + 1681 \)
|
| $53$ |
\( T^{4} - 6 T^{3} + \cdots + 5625 \)
|
| $59$ |
\( T^{4} + 230 T^{2} + 11881 \)
|
| $61$ |
\( T^{4} - 2 T^{3} + \cdots + 35344 \)
|
| $67$ |
\( T^{4} + 12 T^{3} + \cdots + 2601 \)
|
| $71$ |
\( T^{4} - 6 T^{3} + \cdots + 16 \)
|
| $73$ |
\( (T^{2} - 6 T + 12)^{2} \)
|
| $79$ |
\( (T^{2} - 6 T + 36)^{2} \)
|
| $83$ |
\( T^{4} + 62T^{2} + 625 \)
|
| $89$ |
\( T^{4} + 269T^{2} + 2209 \)
|
| $97$ |
\( T^{4} + 39 T^{3} + \cdots + 12321 \)
|
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