Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1089,2,Mod(364,1089)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1089, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1089.364");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1089 = 3^{2} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1089.e (of order \(3\), degree \(2\), 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: | \(8.69570878012\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\zeta_{18})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{3} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | no (minimal twist has level 99) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
| \(\beta_{1}\) | \(=\) |
\( \zeta_{18}^{3} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{18}^{5} + \zeta_{18} \)
|
| \(\beta_{3}\) | \(=\) |
\( -\zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} \)
|
| \(\beta_{4}\) | \(=\) |
\( -\zeta_{18}^{5} + \zeta_{18}^{4} \)
|
| \(\beta_{5}\) | \(=\) |
\( -\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18} \)
|
| \(\zeta_{18}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + 2\beta_{2} ) / 3 \)
|
| \(\zeta_{18}^{2}\) | \(=\) |
\( ( -2\beta_{5} + \beta_{4} + 3\beta_{3} - \beta_{2} ) / 3 \)
|
| \(\zeta_{18}^{3}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{18}^{4}\) | \(=\) |
\( ( -\beta_{5} + 2\beta_{4} + \beta_{2} ) / 3 \)
|
| \(\zeta_{18}^{5}\) | \(=\) |
\( ( -\beta_{5} - \beta_{4} + \beta_{2} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).
| \(n\) | \(244\) | \(848\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 364.1 |
|
−0.939693 | − | 1.62760i | −1.70574 | + | 0.300767i | −0.766044 | + | 1.32683i | −1.43969 | + | 2.49362i | 2.09240 | + | 2.49362i | −0.326352 | − | 0.565258i | −0.879385 | 2.81908 | − | 1.02606i | 5.41147 | ||||||||||||||||||||||
| 364.2 | 0.173648 | + | 0.300767i | 1.11334 | + | 1.32683i | 0.939693 | − | 1.62760i | −0.326352 | + | 0.565258i | −0.205737 | + | 0.565258i | 0.266044 | + | 0.460802i | 1.34730 | −0.520945 | + | 2.95442i | −0.226682 | |||||||||||||||||||||||
| 364.3 | 0.766044 | + | 1.32683i | 0.592396 | − | 1.62760i | −0.173648 | + | 0.300767i | 0.266044 | − | 0.460802i | 2.61334 | − | 0.460802i | −1.43969 | − | 2.49362i | 2.53209 | −2.29813 | − | 1.92836i | 0.815207 | |||||||||||||||||||||||
| 727.1 | −0.939693 | + | 1.62760i | −1.70574 | − | 0.300767i | −0.766044 | − | 1.32683i | −1.43969 | − | 2.49362i | 2.09240 | − | 2.49362i | −0.326352 | + | 0.565258i | −0.879385 | 2.81908 | + | 1.02606i | 5.41147 | |||||||||||||||||||||||
| 727.2 | 0.173648 | − | 0.300767i | 1.11334 | − | 1.32683i | 0.939693 | + | 1.62760i | −0.326352 | − | 0.565258i | −0.205737 | − | 0.565258i | 0.266044 | − | 0.460802i | 1.34730 | −0.520945 | − | 2.95442i | −0.226682 | |||||||||||||||||||||||
| 727.3 | 0.766044 | − | 1.32683i | 0.592396 | + | 1.62760i | −0.173648 | − | 0.300767i | 0.266044 | + | 0.460802i | 2.61334 | + | 0.460802i | −1.43969 | + | 2.49362i | 2.53209 | −2.29813 | + | 1.92836i | 0.815207 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1089.2.e.h | 6 | |
| 9.c | even | 3 | 1 | inner | 1089.2.e.h | 6 | |
| 9.c | even | 3 | 1 | 9801.2.a.be | 3 | ||
| 9.d | odd | 6 | 1 | 9801.2.a.bd | 3 | ||
| 11.b | odd | 2 | 1 | 99.2.e.d | ✓ | 6 | |
| 33.d | even | 2 | 1 | 297.2.e.d | 6 | ||
| 99.g | even | 6 | 1 | 297.2.e.d | 6 | ||
| 99.g | even | 6 | 1 | 891.2.a.k | 3 | ||
| 99.h | odd | 6 | 1 | 99.2.e.d | ✓ | 6 | |
| 99.h | odd | 6 | 1 | 891.2.a.l | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 99.2.e.d | ✓ | 6 | 11.b | odd | 2 | 1 | |
| 99.2.e.d | ✓ | 6 | 99.h | odd | 6 | 1 | |
| 297.2.e.d | 6 | 33.d | even | 2 | 1 | ||
| 297.2.e.d | 6 | 99.g | even | 6 | 1 | ||
| 891.2.a.k | 3 | 99.g | even | 6 | 1 | ||
| 891.2.a.l | 3 | 99.h | odd | 6 | 1 | ||
| 1089.2.e.h | 6 | 1.a | even | 1 | 1 | trivial | |
| 1089.2.e.h | 6 | 9.c | even | 3 | 1 | inner | |
| 9801.2.a.bd | 3 | 9.d | odd | 6 | 1 | ||
| 9801.2.a.be | 3 | 9.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1089, [\chi])\):
|
\( T_{2}^{6} + 3T_{2}^{4} - 2T_{2}^{3} + 9T_{2}^{2} - 3T_{2} + 1 \)
|
|
\( T_{5}^{6} + 3T_{5}^{5} + 9T_{5}^{4} + 2T_{5}^{3} + 3T_{5}^{2} + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 3 T^{4} + \cdots + 1 \)
|
| $3$ |
\( T^{6} + 9T^{3} + 27 \)
|
| $5$ |
\( T^{6} + 3 T^{5} + \cdots + 1 \)
|
| $7$ |
\( T^{6} + 3 T^{5} + \cdots + 1 \)
|
| $11$ |
\( T^{6} \)
|
| $13$ |
\( T^{6} + 9 T^{5} + \cdots + 1 \)
|
| $17$ |
\( (T^{3} - 3 T^{2} - 6 T - 1)^{2} \)
|
| $19$ |
\( (T^{3} - 9 T^{2} + 18 T + 9)^{2} \)
|
| $23$ |
\( T^{6} - 6 T^{5} + \cdots + 361 \)
|
| $29$ |
\( T^{6} + 6 T^{5} + \cdots + 45369 \)
|
| $31$ |
\( T^{6} - 9 T^{5} + \cdots + 361 \)
|
| $37$ |
\( (T^{3} + 12 T^{2} + 27 T - 3)^{2} \)
|
| $41$ |
\( T^{6} + 6 T^{5} + \cdots + 218089 \)
|
| $43$ |
\( T^{6} - 3 T^{5} + \cdots + 294849 \)
|
| $47$ |
\( T^{6} + 12 T^{5} + \cdots + 32041 \)
|
| $53$ |
\( (T^{3} - 3 T^{2} - 126 T + 57)^{2} \)
|
| $59$ |
\( T^{6} + 21 T^{5} + \cdots + 218089 \)
|
| $61$ |
\( T^{6} + 21 T^{5} + \cdots + 103041 \)
|
| $67$ |
\( T^{6} + 3 T^{5} + \cdots + 332929 \)
|
| $71$ |
\( (T^{3} - 12 T^{2} + \cdots + 111)^{2} \)
|
| $73$ |
\( (T^{3} - 6 T^{2} + \cdots + 703)^{2} \)
|
| $79$ |
\( T^{6} + 9 T^{5} + \cdots + 361 \)
|
| $83$ |
\( T^{6} + 27 T^{4} + \cdots + 729 \)
|
| $89$ |
\( (T^{3} + 12 T^{2} + \cdots - 408)^{2} \)
|
| $97$ |
\( T^{6} - 3 T^{5} + \cdots + 516961 \)
|
| show more | |
| show less |