Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9,56,Mod(1,9)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 56, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9.1");
S:= CuspForms(chi, 56);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 56 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9.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: | \(172.423320917\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{4} - x^{3} - 147352813458782x^{2} + 335293764586979486388x + 1328731829156322035389049544 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{22}\cdot 3^{11}\cdot 5^{2}\cdot 7\cdot 11 \) |
| Twist minimal: | no (minimal twist has level 3) |
| 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
\( x^{4} - x^{3} - 147352813458782x^{2} + 335293764586979486388x + 1328731829156322035389049544 \)
:
| \(\beta_{1}\) | \(=\) |
\( 24\nu - 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 27\nu^{3} + 93639087\nu^{2} - 3171982698819144\nu - 109294917593315951988 ) / 30592 \)
|
| \(\beta_{3}\) | \(=\) |
\( 576\nu^{2} + 1965986928\nu - 42437610767626092 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 6 ) / 24 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 81916122\beta _1 + 42437610276129360 ) / 576 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -10404343\beta_{3} + 1957888\beta_{2} + 9310903960702230\beta _1 - 434540527935479171776944 ) / 1728 \)
|
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 |
|
−2.51419e8 | 0 | 2.71827e16 | 8.80534e18 | 0 | 4.60331e21 | 2.22409e24 | 0 | −2.21383e27 | ||||||||||||||||||||||||||||||
| 1.2 | 6.98234e6 | 0 | −3.59800e16 | −1.79430e19 | 0 | −5.98408e22 | −5.02790e23 | 0 | −1.25284e26 | |||||||||||||||||||||||||||||||
| 1.3 | 1.75215e8 | 0 | −5.32837e15 | 3.05713e19 | 0 | −2.55745e23 | −7.24641e24 | 0 | 5.35657e27 | |||||||||||||||||||||||||||||||
| 1.4 | 2.98147e8 | 0 | 5.28627e16 | −5.33679e18 | 0 | 1.81847e23 | 5.01899e24 | 0 | −1.59115e27 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
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 | 9.56.a.b | 4 | |
| 3.b | odd | 2 | 1 | 3.56.a.a | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.56.a.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 9.56.a.b | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 228925656 T_{2}^{3} + \cdots - 91\!\cdots\!16 \)
acting on \(S_{56}^{\mathrm{new}}(\Gamma_0(9))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + \cdots - 91\!\cdots\!16 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 25\!\cdots\!00 \)
|
| $7$ |
\( T^{4} + \cdots + 12\!\cdots\!00 \)
|
| $11$ |
\( T^{4} + \cdots + 23\!\cdots\!24 \)
|
| $13$ |
\( T^{4} + \cdots + 10\!\cdots\!04 \)
|
| $17$ |
\( T^{4} + \cdots + 75\!\cdots\!56 \)
|
| $19$ |
\( T^{4} + \cdots - 84\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots + 17\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots + 30\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots - 18\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots - 96\!\cdots\!00 \)
|
| $41$ |
\( T^{4} + \cdots - 27\!\cdots\!00 \)
|
| $43$ |
\( T^{4} + \cdots + 54\!\cdots\!16 \)
|
| $47$ |
\( T^{4} + \cdots - 10\!\cdots\!84 \)
|
| $53$ |
\( T^{4} + \cdots + 16\!\cdots\!00 \)
|
| $59$ |
\( T^{4} + \cdots - 11\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots - 72\!\cdots\!04 \)
|
| $67$ |
\( T^{4} + \cdots + 42\!\cdots\!76 \)
|
| $71$ |
\( T^{4} + \cdots + 13\!\cdots\!16 \)
|
| $73$ |
\( T^{4} + \cdots - 43\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots + 91\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots - 45\!\cdots\!96 \)
|
| $89$ |
\( T^{4} + \cdots + 24\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots - 14\!\cdots\!64 \)
|
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