Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9,36,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, 36, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9.1");
S:= CuspForms(chi, 36);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 36 \) |
| 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: | \(69.8356175703\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 12422194x - 2645665785 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3^{5}\cdot 5\cdot 7 \) |
| Twist minimal: | no (minimal twist has level 1) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - 12422194x - 2645665785 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 24\nu^{2} + 44712\nu - 198755104 ) / 979 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -352344\nu^{2} + 764621208\nu + 2917923681824 ) / 979 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 14681\beta_1 ) / 1451520 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -69\beta_{2} + 1179971\beta _1 + 445211432960 ) / 53760 \)
|
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 |
|
−331573. | 0 | 7.55810e10 | −2.12139e12 | 0 | 3.96640e14 | −1.36679e16 | 0 | 7.03395e17 | |||||||||||||||||||||||||||
| 1.2 | 26808.0 | 0 | −3.36411e10 | −8.21401e11 | 0 | 6.06942e14 | −1.82297e15 | 0 | −2.20201e16 | ||||||||||||||||||||||||||||
| 1.3 | 165109. | 0 | −7.09870e9 | 2.05014e12 | 0 | −1.25160e14 | −6.84517e15 | 0 | 3.38496e17 | ||||||||||||||||||||||||||||
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 | 9.36.a.b | 3 | |
| 3.b | odd | 2 | 1 | 1.36.a.a | ✓ | 3 | |
| 12.b | even | 2 | 1 | 16.36.a.d | 3 | ||
| 15.d | odd | 2 | 1 | 25.36.a.a | 3 | ||
| 15.e | even | 4 | 2 | 25.36.b.a | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.36.a.a | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 9.36.a.b | 3 | 1.a | even | 1 | 1 | trivial | |
| 16.36.a.d | 3 | 12.b | even | 2 | 1 | ||
| 25.36.a.a | 3 | 15.d | odd | 2 | 1 | ||
| 25.36.b.a | 6 | 15.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} + 139656T_{2}^{2} - 59208339456T_{2} + 1467625047588864 \)
acting on \(S_{36}^{\mathrm{new}}(\Gamma_0(9))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + \cdots + 14\!\cdots\!64 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + \cdots - 35\!\cdots\!00 \)
|
| $7$ |
\( T^{3} + \cdots + 30\!\cdots\!16 \)
|
| $11$ |
\( T^{3} + \cdots + 14\!\cdots\!08 \)
|
| $13$ |
\( T^{3} + \cdots + 49\!\cdots\!48 \)
|
| $17$ |
\( T^{3} + \cdots + 68\!\cdots\!24 \)
|
| $19$ |
\( T^{3} + \cdots - 49\!\cdots\!00 \)
|
| $23$ |
\( T^{3} + \cdots + 20\!\cdots\!72 \)
|
| $29$ |
\( T^{3} + \cdots + 25\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots + 11\!\cdots\!92 \)
|
| $37$ |
\( T^{3} + \cdots + 36\!\cdots\!96 \)
|
| $41$ |
\( T^{3} + \cdots - 11\!\cdots\!92 \)
|
| $43$ |
\( T^{3} + \cdots - 12\!\cdots\!12 \)
|
| $47$ |
\( T^{3} + \cdots - 47\!\cdots\!56 \)
|
| $53$ |
\( T^{3} + \cdots + 44\!\cdots\!32 \)
|
| $59$ |
\( T^{3} + \cdots - 10\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots + 57\!\cdots\!92 \)
|
| $67$ |
\( T^{3} + \cdots - 10\!\cdots\!24 \)
|
| $71$ |
\( T^{3} + \cdots - 33\!\cdots\!92 \)
|
| $73$ |
\( T^{3} + \cdots + 22\!\cdots\!28 \)
|
| $79$ |
\( T^{3} + \cdots - 88\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots + 11\!\cdots\!92 \)
|
| $89$ |
\( T^{3} + \cdots + 91\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots - 83\!\cdots\!44 \)
|
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