Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [686,2,Mod(1,686)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(686, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("686.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 686 = 2 \cdot 7^{3} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 686.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: | \(5.47773757866\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.10910144.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 13x^{4} + 54x^{2} - 71 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 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
\( x^{6} - 13x^{4} + 54x^{2} - 71 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 9\nu^{2} + 19 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 9\nu^{3} + 19\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 9\beta_{2} + 17 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 9\beta_{3} + 26\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
1.00000 | −2.40872 | 1.00000 | −1.07198 | −2.40872 | 0 | 1.00000 | 2.80194 | −1.07198 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.10833 | 1.00000 | 2.62904 | −2.10833 | 0 | 1.00000 | 1.44504 | 2.62904 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.65922 | 1.00000 | −2.98982 | −1.65922 | 0 | 1.00000 | −0.246980 | −2.98982 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 1.65922 | 1.00000 | 2.98982 | 1.65922 | 0 | 1.00000 | −0.246980 | 2.98982 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 2.10833 | 1.00000 | −2.62904 | 2.10833 | 0 | 1.00000 | 1.44504 | −2.62904 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 2.40872 | 1.00000 | 1.07198 | 2.40872 | 0 | 1.00000 | 2.80194 | 1.07198 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(7\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 686.2.a.f | ✓ | 6 |
| 3.b | odd | 2 | 1 | 6174.2.a.s | 6 | ||
| 4.b | odd | 2 | 1 | 5488.2.a.m | 6 | ||
| 7.b | odd | 2 | 1 | inner | 686.2.a.f | ✓ | 6 |
| 7.c | even | 3 | 2 | 686.2.c.e | 12 | ||
| 7.d | odd | 6 | 2 | 686.2.c.e | 12 | ||
| 21.c | even | 2 | 1 | 6174.2.a.s | 6 | ||
| 28.d | even | 2 | 1 | 5488.2.a.m | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 686.2.a.f | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 686.2.a.f | ✓ | 6 | 7.b | odd | 2 | 1 | inner |
| 686.2.c.e | 12 | 7.c | even | 3 | 2 | ||
| 686.2.c.e | 12 | 7.d | odd | 6 | 2 | ||
| 5488.2.a.m | 6 | 4.b | odd | 2 | 1 | ||
| 5488.2.a.m | 6 | 28.d | even | 2 | 1 | ||
| 6174.2.a.s | 6 | 3.b | odd | 2 | 1 | ||
| 6174.2.a.s | 6 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 13T_{3}^{4} + 54T_{3}^{2} - 71 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(686))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
|
| $3$ |
\( T^{6} - 13 T^{4} + \cdots - 71 \)
|
| $5$ |
\( T^{6} - 17 T^{4} + \cdots - 71 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( (T^{3} - 8 T^{2} + 12 T + 8)^{2} \)
|
| $13$ |
\( T^{6} - 63 T^{4} + \cdots - 3479 \)
|
| $17$ |
\( T^{6} - 76 T^{4} + \cdots - 4544 \)
|
| $19$ |
\( T^{6} - 52 T^{4} + \cdots - 4544 \)
|
| $23$ |
\( (T^{3} - 9 T^{2} + \cdots + 211)^{2} \)
|
| $29$ |
\( (T^{3} - 6 T^{2} - 16 T - 8)^{2} \)
|
| $31$ |
\( T^{6} - 132 T^{4} + \cdots - 4544 \)
|
| $37$ |
\( (T^{3} - 2 T^{2} - 8 T + 8)^{2} \)
|
| $41$ |
\( T^{6} - 196 T^{4} + \cdots - 222656 \)
|
| $43$ |
\( (T^{3} - 6 T^{2} + \cdots + 104)^{2} \)
|
| $47$ |
\( T^{6} - 276 T^{4} + \cdots - 767936 \)
|
| $53$ |
\( (T^{3} + 6 T^{2} + \cdots - 104)^{2} \)
|
| $59$ |
\( T^{6} - 391 T^{4} + \cdots - 2027831 \)
|
| $61$ |
\( T^{6} - 171 T^{4} + \cdots - 51759 \)
|
| $67$ |
\( (T^{3} + 6 T^{2} + \cdots - 216)^{2} \)
|
| $71$ |
\( (T^{3} + T^{2} - 184 T - 911)^{2} \)
|
| $73$ |
\( T^{6} - 244 T^{4} + \cdots - 4544 \)
|
| $79$ |
\( (T^{3} - 9 T^{2} - 64 T - 41)^{2} \)
|
| $83$ |
\( T^{6} - 77 T^{4} + \cdots - 3479 \)
|
| $89$ |
\( T^{6} - 304 T^{4} + \cdots - 290816 \)
|
| $97$ |
\( T^{6} - 252 T^{4} + \cdots - 222656 \)
|
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