Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9,22,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, 22, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9.1");
S:= CuspForms(chi, 22);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| 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: | \(25.1529609858\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{649}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 162 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2\cdot 3^{2}\cdot 7 \) |
| 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 \(\beta = 63\sqrt{649}\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
−1937.96 | 0 | 1.65852e6 | −2.22745e7 | 0 | 4.78205e7 | 8.50053e8 | 0 | 4.31669e10 | ||||||||||||||||||||||||
| 1.2 | 1271.96 | 0 | −479282. | 2.12776e7 | 0 | 6.32076e8 | −3.27711e9 | 0 | 2.70641e10 | |||||||||||||||||||||||||
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 | 9.22.a.e | 2 | |
| 3.b | odd | 2 | 1 | 3.22.a.c | ✓ | 2 | |
| 12.b | even | 2 | 1 | 48.22.a.g | 2 | ||
| 15.d | odd | 2 | 1 | 75.22.a.d | 2 | ||
| 15.e | even | 4 | 2 | 75.22.b.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.22.a.c | ✓ | 2 | 3.b | odd | 2 | 1 | |
| 9.22.a.e | 2 | 1.a | even | 1 | 1 | trivial | |
| 48.22.a.g | 2 | 12.b | even | 2 | 1 | ||
| 75.22.a.d | 2 | 15.d | odd | 2 | 1 | ||
| 75.22.b.d | 4 | 15.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 666T_{2} - 2464992 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(9))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 666 T - 2464992 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} + \cdots - 473947100199900 \)
|
| $7$ |
\( T^{2} + \cdots + 30\!\cdots\!00 \)
|
| $11$ |
\( T^{2} + \cdots + 95\!\cdots\!12 \)
|
| $13$ |
\( T^{2} + \cdots - 58\!\cdots\!92 \)
|
| $17$ |
\( T^{2} + \cdots + 24\!\cdots\!24 \)
|
| $19$ |
\( T^{2} + \cdots - 12\!\cdots\!20 \)
|
| $23$ |
\( T^{2} + \cdots - 85\!\cdots\!00 \)
|
| $29$ |
\( T^{2} + \cdots - 26\!\cdots\!00 \)
|
| $31$ |
\( T^{2} + \cdots + 23\!\cdots\!00 \)
|
| $37$ |
\( T^{2} + \cdots - 54\!\cdots\!20 \)
|
| $41$ |
\( T^{2} + \cdots - 32\!\cdots\!80 \)
|
| $43$ |
\( T^{2} + \cdots + 20\!\cdots\!44 \)
|
| $47$ |
\( T^{2} + \cdots + 17\!\cdots\!16 \)
|
| $53$ |
\( T^{2} + \cdots - 20\!\cdots\!60 \)
|
| $59$ |
\( T^{2} + \cdots - 83\!\cdots\!20 \)
|
| $61$ |
\( T^{2} + \cdots - 74\!\cdots\!84 \)
|
| $67$ |
\( T^{2} + \cdots + 38\!\cdots\!04 \)
|
| $71$ |
\( T^{2} + \cdots + 73\!\cdots\!84 \)
|
| $73$ |
\( T^{2} + \cdots - 11\!\cdots\!40 \)
|
| $79$ |
\( T^{2} + \cdots + 69\!\cdots\!00 \)
|
| $83$ |
\( T^{2} + \cdots + 33\!\cdots\!12 \)
|
| $89$ |
\( T^{2} + \cdots - 17\!\cdots\!80 \)
|
| $97$ |
\( T^{2} + \cdots - 27\!\cdots\!76 \)
|
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