Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [688,2,Mod(687,688)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(688, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("688.687");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 688 = 2^{4} \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 688.g (of order \(2\), degree \(1\), 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: | \(5.49370765906\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-43})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 10x^{2} - 11x + 121 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{43}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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 in terms of a root \(\nu\) of
\( x^{4} - x^{3} - 10x^{2} - 11x + 121 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 10\nu^{2} - 10\nu - 66 ) / 55 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + \nu^{2} + 21\nu + 11 ) / 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -6\nu^{3} - 5\nu^{2} + 5\nu + 121 ) / 55 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 11\beta _1 + 10 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -10\beta_{3} - 5\beta _1 + 16 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/688\mathbb{Z}\right)^\times\).
| \(n\) | \(431\) | \(433\) | \(517\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 687.1 |
|
0 | 0 | 0 | 0 | 0 | 0 | 0 | −3.00000 | 0 | ||||||||||||||||||||||||||||||
| 687.2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −3.00000 | 0 | |||||||||||||||||||||||||||||||
| 687.3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −3.00000 | 0 | |||||||||||||||||||||||||||||||
| 687.4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −3.00000 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 43.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-43}) \) |
| 4.b | odd | 2 | 1 | inner |
| 172.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 688.2.g.d | ✓ | 4 |
| 4.b | odd | 2 | 1 | inner | 688.2.g.d | ✓ | 4 |
| 8.b | even | 2 | 1 | 2752.2.g.d | 4 | ||
| 8.d | odd | 2 | 1 | 2752.2.g.d | 4 | ||
| 43.b | odd | 2 | 1 | CM | 688.2.g.d | ✓ | 4 |
| 172.d | even | 2 | 1 | inner | 688.2.g.d | ✓ | 4 |
| 344.e | even | 2 | 1 | 2752.2.g.d | 4 | ||
| 344.h | odd | 2 | 1 | 2752.2.g.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 688.2.g.d | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 688.2.g.d | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
| 688.2.g.d | ✓ | 4 | 43.b | odd | 2 | 1 | CM |
| 688.2.g.d | ✓ | 4 | 172.d | even | 2 | 1 | inner |
| 2752.2.g.d | 4 | 8.b | even | 2 | 1 | ||
| 2752.2.g.d | 4 | 8.d | odd | 2 | 1 | ||
| 2752.2.g.d | 4 | 344.e | even | 2 | 1 | ||
| 2752.2.g.d | 4 | 344.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(688, [\chi])\):
|
\( T_{3} \)
|
|
\( T_{11}^{4} + 23T_{11}^{2} + 100 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + 23T^{2} + 100 \)
|
| $13$ |
\( (T^{2} + 3 T - 30)^{2} \)
|
| $17$ |
\( (T^{2} - 5 T - 26)^{2} \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( T^{4} + 95T^{2} + 676 \)
|
| $29$ |
\( T^{4} \)
|
| $31$ |
\( T^{4} + 143T^{2} + 2500 \)
|
| $37$ |
\( T^{4} \)
|
| $41$ |
\( (T^{2} + 11 T - 2)^{2} \)
|
| $43$ |
\( (T^{2} + 43)^{2} \)
|
| $47$ |
\( (T^{2} + 172)^{2} \)
|
| $53$ |
\( (T^{2} - 13 T + 10)^{2} \)
|
| $59$ |
\( (T^{2} + 172)^{2} \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( T^{4} + 359 T^{2} + 24964 \)
|
| $71$ |
\( T^{4} \)
|
| $73$ |
\( T^{4} \)
|
| $79$ |
\( (T^{2} + 172)^{2} \)
|
| $83$ |
\( T^{4} + 455 T^{2} + 42436 \)
|
| $89$ |
\( T^{4} \)
|
| $97$ |
\( (T^{2} - T - 290)^{2} \)
|
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