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: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/688\mathbb{Z}\right)^\times\).
| \(n\) | \(431\) | \(433\) | \(517\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | 2.00000 | 0 | − | 3.46410i | 0 | 2.00000 | 0 | 1.00000 | 0 | |||||||||||||||||||||||
| 687.2 | 0 | 2.00000 | 0 | 3.46410i | 0 | 2.00000 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 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.c | yes | 2 |
| 4.b | odd | 2 | 1 | 688.2.g.a | ✓ | 2 | |
| 8.b | even | 2 | 1 | 2752.2.g.a | 2 | ||
| 8.d | odd | 2 | 1 | 2752.2.g.c | 2 | ||
| 43.b | odd | 2 | 1 | 688.2.g.a | ✓ | 2 | |
| 172.d | even | 2 | 1 | inner | 688.2.g.c | yes | 2 |
| 344.e | even | 2 | 1 | 2752.2.g.a | 2 | ||
| 344.h | odd | 2 | 1 | 2752.2.g.c | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 688.2.g.a | ✓ | 2 | 4.b | odd | 2 | 1 | |
| 688.2.g.a | ✓ | 2 | 43.b | odd | 2 | 1 | |
| 688.2.g.c | yes | 2 | 1.a | even | 1 | 1 | trivial |
| 688.2.g.c | yes | 2 | 172.d | even | 2 | 1 | inner |
| 2752.2.g.a | 2 | 8.b | even | 2 | 1 | ||
| 2752.2.g.a | 2 | 344.e | even | 2 | 1 | ||
| 2752.2.g.c | 2 | 8.d | odd | 2 | 1 | ||
| 2752.2.g.c | 2 | 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 \(S_{2}^{\mathrm{new}}(688, [\chi])\):
|
\( T_{3} - 2 \)
|
|
\( T_{11}^{2} + 3 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( (T - 2)^{2} \)
|
| $5$ |
\( T^{2} + 12 \)
|
| $7$ |
\( (T - 2)^{2} \)
|
| $11$ |
\( T^{2} + 3 \)
|
| $13$ |
\( (T + 1)^{2} \)
|
| $17$ |
\( (T - 3)^{2} \)
|
| $19$ |
\( (T - 4)^{2} \)
|
| $23$ |
\( T^{2} + 27 \)
|
| $29$ |
\( T^{2} + 12 \)
|
| $31$ |
\( T^{2} + 27 \)
|
| $37$ |
\( T^{2} \)
|
| $41$ |
\( (T + 9)^{2} \)
|
| $43$ |
\( T^{2} - 8T + 43 \)
|
| $47$ |
\( T^{2} + 12 \)
|
| $53$ |
\( (T - 3)^{2} \)
|
| $59$ |
\( T^{2} + 108 \)
|
| $61$ |
\( T^{2} + 108 \)
|
| $67$ |
\( T^{2} + 243 \)
|
| $71$ |
\( (T - 12)^{2} \)
|
| $73$ |
\( T^{2} + 108 \)
|
| $79$ |
\( T^{2} + 108 \)
|
| $83$ |
\( T^{2} + 75 \)
|
| $89$ |
\( T^{2} + 108 \)
|
| $97$ |
\( (T - 7)^{2} \)
|
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