Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9,30,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, 30, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9.1");
S:= CuspForms(chi, 30);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 30 \) |
| 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: | \(47.9502381447\) |
| 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: |
\( x^{4} - 7868915x^{2} + 1537952416000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3^{10}\cdot 5^{3}\cdot 7 \) |
| 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
\( x^{4} - 7868915x^{2} + 1537952416000 \)
:
| \(\beta_{1}\) | \(=\) |
\( 12\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 108\nu^{3} - 807626340\nu ) / 19 \)
|
| \(\beta_{3}\) | \(=\) |
\( 144\nu^{2} - 566561880 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 566561880 ) / 144 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 19\beta_{2} + 67302195\beta_1 ) / 108 \)
|
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 |
|
−33230.2 | 0 | 5.67372e8 | 3.33626e9 | 0 | −3.17729e10 | −1.01357e12 | 0 | −1.10865e14 | ||||||||||||||||||||||||||||||
| 1.2 | −5374.05 | 0 | −5.07991e8 | −1.84705e10 | 0 | 2.84475e10 | 5.61514e12 | 0 | 9.92613e13 | |||||||||||||||||||||||||||||||
| 1.3 | 5374.05 | 0 | −5.07991e8 | 1.84705e10 | 0 | 2.84475e10 | −5.61514e12 | 0 | 9.92613e13 | |||||||||||||||||||||||||||||||
| 1.4 | 33230.2 | 0 | 5.67372e8 | −3.33626e9 | 0 | −3.17729e10 | 1.01357e12 | 0 | −1.10865e14 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 9.30.a.d | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 9.30.a.d | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 9.30.a.d | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 9.30.a.d | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 1133123760T_{2}^{2} + 31890981298176000 \)
acting on \(S_{30}^{\mathrm{new}}(\Gamma_0(9))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + \cdots + 31\!\cdots\!00 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 37\!\cdots\!00 \)
|
| $7$ |
\( (T^{2} + \cdots - 90\!\cdots\!00)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 89\!\cdots\!00 \)
|
| $13$ |
\( (T^{2} + \cdots - 29\!\cdots\!00)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 28\!\cdots\!00 \)
|
| $19$ |
\( (T^{2} + \cdots - 80\!\cdots\!44)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 13\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots + 42\!\cdots\!00 \)
|
| $31$ |
\( (T^{2} + \cdots - 12\!\cdots\!76)^{2} \)
|
| $37$ |
\( (T^{2} + \cdots - 81\!\cdots\!00)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 38\!\cdots\!00 \)
|
| $43$ |
\( (T^{2} + \cdots - 99\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 20\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots + 12\!\cdots\!00 \)
|
| $59$ |
\( T^{4} + \cdots + 69\!\cdots\!00 \)
|
| $61$ |
\( (T^{2} + \cdots + 84\!\cdots\!24)^{2} \)
|
| $67$ |
\( (T^{2} + \cdots - 48\!\cdots\!00)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 38\!\cdots\!00 \)
|
| $73$ |
\( (T^{2} + \cdots - 49\!\cdots\!00)^{2} \)
|
| $79$ |
\( (T^{2} + \cdots + 10\!\cdots\!16)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 82\!\cdots\!00 \)
|
| $89$ |
\( T^{4} + \cdots + 13\!\cdots\!00 \)
|
| $97$ |
\( (T^{2} + \cdots - 79\!\cdots\!00)^{2} \)
|
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