Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [688,2,Mod(1,688)]
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([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("688.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 688 = 2^{4} \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 688.a (trivial) |
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: | yes |
| Analytic conductor: | \(5.49370765906\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{5}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 86) |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −1.61803 | 0 | −2.61803 | 0 | 4.47214 | 0 | −0.381966 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 0.618034 | 0 | −0.381966 | 0 | −4.47214 | 0 | −2.61803 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(43\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 688.2.a.e | 2 | |
| 3.b | odd | 2 | 1 | 6192.2.a.bp | 2 | ||
| 4.b | odd | 2 | 1 | 86.2.a.b | ✓ | 2 | |
| 8.b | even | 2 | 1 | 2752.2.a.p | 2 | ||
| 8.d | odd | 2 | 1 | 2752.2.a.k | 2 | ||
| 12.b | even | 2 | 1 | 774.2.a.k | 2 | ||
| 20.d | odd | 2 | 1 | 2150.2.a.t | 2 | ||
| 20.e | even | 4 | 2 | 2150.2.b.q | 4 | ||
| 28.d | even | 2 | 1 | 4214.2.a.j | 2 | ||
| 172.d | even | 2 | 1 | 3698.2.a.e | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 86.2.a.b | ✓ | 2 | 4.b | odd | 2 | 1 | |
| 688.2.a.e | 2 | 1.a | even | 1 | 1 | trivial | |
| 774.2.a.k | 2 | 12.b | even | 2 | 1 | ||
| 2150.2.a.t | 2 | 20.d | odd | 2 | 1 | ||
| 2150.2.b.q | 4 | 20.e | even | 4 | 2 | ||
| 2752.2.a.k | 2 | 8.d | odd | 2 | 1 | ||
| 2752.2.a.p | 2 | 8.b | even | 2 | 1 | ||
| 3698.2.a.e | 2 | 172.d | even | 2 | 1 | ||
| 4214.2.a.j | 2 | 28.d | even | 2 | 1 | ||
| 6192.2.a.bp | 2 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(688))\):
|
\( T_{3}^{2} + T_{3} - 1 \)
|
|
\( T_{5}^{2} + 3T_{5} + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + T - 1 \)
|
| $5$ |
\( T^{2} + 3T + 1 \)
|
| $7$ |
\( T^{2} - 20 \)
|
| $11$ |
\( T^{2} - 4T - 16 \)
|
| $13$ |
\( T^{2} - 20 \)
|
| $17$ |
\( T^{2} + T - 1 \)
|
| $19$ |
\( T^{2} + 11T + 29 \)
|
| $23$ |
\( T^{2} + 3T - 9 \)
|
| $29$ |
\( T^{2} + 7T + 1 \)
|
| $31$ |
\( T^{2} + 13T + 41 \)
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| $37$ |
\( T^{2} + 5T + 5 \)
|
| $41$ |
\( T^{2} + 5T - 5 \)
|
| $43$ |
\( (T - 1)^{2} \)
|
| $47$ |
\( T^{2} + 3T - 59 \)
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| $53$ |
\( T^{2} + 10T + 20 \)
|
| $59$ |
\( T^{2} + 16T + 44 \)
|
| $61$ |
\( T^{2} - 4T - 76 \)
|
| $67$ |
\( (T + 2)^{2} \)
|
| $71$ |
\( T^{2} - 16T + 44 \)
|
| $73$ |
\( T^{2} - 4T - 76 \)
|
| $79$ |
\( T^{2} - T - 1 \)
|
| $83$ |
\( T^{2} - 10T - 20 \)
|
| $89$ |
\( T^{2} - 2T - 44 \)
|
| $97$ |
\( T^{2} + 11T - 1 \)
|
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