Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9,9,Mod(8,9)]
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([1]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9.8");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9.b (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(3.66640749055\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-2}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} + 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(\beta = 3\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/9\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 8.1 |
|
− | 4.24264i | 0 | 238.000 | − | 988.535i | 0 | 1652.00 | − | 2095.86i | 0 | −4194.00 | |||||||||||||||||||||
| 8.2 | 4.24264i | 0 | 238.000 | 988.535i | 0 | 1652.00 | 2095.86i | 0 | −4194.00 | |||||||||||||||||||||||||
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 | 9.9.b.a | ✓ | 2 |
| 3.b | odd | 2 | 1 | inner | 9.9.b.a | ✓ | 2 |
| 4.b | odd | 2 | 1 | 144.9.e.a | 2 | ||
| 5.b | even | 2 | 1 | 225.9.c.a | 2 | ||
| 5.c | odd | 4 | 2 | 225.9.d.a | 4 | ||
| 9.c | even | 3 | 2 | 81.9.d.b | 4 | ||
| 9.d | odd | 6 | 2 | 81.9.d.b | 4 | ||
| 12.b | even | 2 | 1 | 144.9.e.a | 2 | ||
| 15.d | odd | 2 | 1 | 225.9.c.a | 2 | ||
| 15.e | even | 4 | 2 | 225.9.d.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 9.9.b.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 9.9.b.a | ✓ | 2 | 3.b | odd | 2 | 1 | inner |
| 81.9.d.b | 4 | 9.c | even | 3 | 2 | ||
| 81.9.d.b | 4 | 9.d | odd | 6 | 2 | ||
| 144.9.e.a | 2 | 4.b | odd | 2 | 1 | ||
| 144.9.e.a | 2 | 12.b | even | 2 | 1 | ||
| 225.9.c.a | 2 | 5.b | even | 2 | 1 | ||
| 225.9.c.a | 2 | 15.d | odd | 2 | 1 | ||
| 225.9.d.a | 4 | 5.c | odd | 4 | 2 | ||
| 225.9.d.a | 4 | 15.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(9, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 18 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} + 977202 \)
|
| $7$ |
\( (T - 1652)^{2} \)
|
| $11$ |
\( T^{2} + 456503328 \)
|
| $13$ |
\( (T + 26272)^{2} \)
|
| $17$ |
\( T^{2} + 230566338 \)
|
| $19$ |
\( (T - 46640)^{2} \)
|
| $23$ |
\( T^{2} + 107644288032 \)
|
| $29$ |
\( T^{2} + 378204699762 \)
|
| $31$ |
\( (T + 196444)^{2} \)
|
| $37$ |
\( (T - 2819414)^{2} \)
|
| $41$ |
\( T^{2} + 499042458882 \)
|
| $43$ |
\( (T + 2213464)^{2} \)
|
| $47$ |
\( T^{2} + 2666553329952 \)
|
| $53$ |
\( T^{2} + 26922373529202 \)
|
| $59$ |
\( T^{2} + 137939140326528 \)
|
| $61$ |
\( (T + 17405302)^{2} \)
|
| $67$ |
\( (T + 14322664)^{2} \)
|
| $71$ |
\( T^{2} + 244785425278752 \)
|
| $73$ |
\( (T + 8906992)^{2} \)
|
| $79$ |
\( (T - 32758844)^{2} \)
|
| $83$ |
\( T^{2} + 72\!\cdots\!12 \)
|
| $89$ |
\( T^{2} + 34\!\cdots\!38 \)
|
| $97$ |
\( (T + 24451744)^{2} \)
|
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