Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [56,10,Mod(1,56)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(56, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("56.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 56 = 2^{3} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 56.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: | \(28.8420068252\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 63x^{2} - 176x + 28 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{13}\cdot 3^{3}\cdot 17 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 63x^{2} - 176x + 28 \)
:
| \(\beta_{1}\) | \(=\) |
\( -6\nu^{2} + 48\nu + 189 \)
|
| \(\beta_{2}\) | \(=\) |
\( -32\nu^{3} + 70\nu^{2} + 1552\nu + 2019 \)
|
| \(\beta_{3}\) | \(=\) |
\( -288\nu^{3} + 1332\nu^{2} + 13248\nu - 3942 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - 9\beta_{2} + 117\beta_1 ) / 4896 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 9\beta_{2} + 15\beta _1 + 19278 ) / 612 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 22\beta_{3} - 249\beta_{2} + 1979\beta _1 + 215424 ) / 1632 \)
|
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 | −175.108 | 0 | 2247.52 | 0 | −2401.00 | 0 | 10979.9 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −110.374 | 0 | −1849.79 | 0 | −2401.00 | 0 | −7500.49 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 96.1756 | 0 | −594.465 | 0 | −2401.00 | 0 | −10433.3 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 273.307 | 0 | 1736.73 | 0 | −2401.00 | 0 | 55013.8 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(7\) | \( +1 \) |
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 | 56.10.a.d | ✓ | 4 |
| 4.b | odd | 2 | 1 | 112.10.a.j | 4 | ||
| 7.b | odd | 2 | 1 | 392.10.a.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 56.10.a.d | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 112.10.a.j | 4 | 4.b | odd | 2 | 1 | ||
| 392.10.a.e | 4 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 84T_{3}^{3} - 59868T_{3}^{2} + 362880T_{3} + 508032000 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(56))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 84 T^{3} + \cdots + 508032000 \)
|
| $5$ |
\( T^{4} + \cdots + 4292242257920 \)
|
| $7$ |
\( (T + 2401)^{4} \)
|
| $11$ |
\( T^{4} + \cdots - 27\!\cdots\!76 \)
|
| $13$ |
\( T^{4} + \cdots - 85\!\cdots\!64 \)
|
| $17$ |
\( T^{4} + \cdots - 66\!\cdots\!72 \)
|
| $19$ |
\( T^{4} + \cdots - 22\!\cdots\!68 \)
|
| $23$ |
\( T^{4} + \cdots + 79\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots + 27\!\cdots\!04 \)
|
| $31$ |
\( T^{4} + \cdots + 37\!\cdots\!56 \)
|
| $37$ |
\( T^{4} + \cdots - 25\!\cdots\!84 \)
|
| $41$ |
\( T^{4} + \cdots + 11\!\cdots\!72 \)
|
| $43$ |
\( T^{4} + \cdots - 56\!\cdots\!44 \)
|
| $47$ |
\( T^{4} + \cdots + 16\!\cdots\!12 \)
|
| $53$ |
\( T^{4} + \cdots - 80\!\cdots\!48 \)
|
| $59$ |
\( T^{4} + \cdots + 10\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots - 17\!\cdots\!24 \)
|
| $67$ |
\( T^{4} + \cdots - 10\!\cdots\!04 \)
|
| $71$ |
\( T^{4} + \cdots + 99\!\cdots\!76 \)
|
| $73$ |
\( T^{4} + \cdots + 60\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots - 16\!\cdots\!12 \)
|
| $83$ |
\( T^{4} + \cdots - 12\!\cdots\!52 \)
|
| $89$ |
\( T^{4} + \cdots + 28\!\cdots\!28 \)
|
| $97$ |
\( T^{4} + \cdots - 46\!\cdots\!76 \)
|
| show more | |
| show less |