Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [729,2,Mod(82,729)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(729, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([8]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("729.82");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 729 = 3^{6} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 729.e (of order \(9\), degree \(6\), not 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: | \(5.82109430735\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| 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: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 243) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-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 primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/729\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-\zeta_{18}^{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 82.1 |
|
−0.439693 | − | 2.49362i | 0 | −4.14543 | + | 1.50881i | −0.358441 | − | 0.300767i | 0 | 3.03209 | + | 1.10359i | 3.05303 | + | 5.28801i | 0 | −0.592396 | + | 1.02606i | ||||||||||||||||||||||||
| 163.1 | 1.26604 | − | 0.460802i | 0 | −0.141559 | + | 0.118782i | −0.286989 | − | 1.62760i | 0 | 1.84730 | + | 1.55007i | −1.47178 | + | 2.54920i | 0 | −1.11334 | − | 1.92836i | |||||||||||||||||||||||||
| 325.1 | 0.673648 | − | 0.565258i | 0 | −0.213011 | + | 1.20805i | 3.64543 | − | 1.32683i | 0 | −0.379385 | − | 2.15160i | 1.41875 | + | 2.45734i | 0 | 1.70574 | − | 2.95442i | |||||||||||||||||||||||||
| 406.1 | 0.673648 | + | 0.565258i | 0 | −0.213011 | − | 1.20805i | 3.64543 | + | 1.32683i | 0 | −0.379385 | + | 2.15160i | 1.41875 | − | 2.45734i | 0 | 1.70574 | + | 2.95442i | |||||||||||||||||||||||||
| 568.1 | 1.26604 | + | 0.460802i | 0 | −0.141559 | − | 0.118782i | −0.286989 | + | 1.62760i | 0 | 1.84730 | − | 1.55007i | −1.47178 | − | 2.54920i | 0 | −1.11334 | + | 1.92836i | |||||||||||||||||||||||||
| 649.1 | −0.439693 | + | 2.49362i | 0 | −4.14543 | − | 1.50881i | −0.358441 | + | 0.300767i | 0 | 3.03209 | − | 1.10359i | 3.05303 | − | 5.28801i | 0 | −0.592396 | − | 1.02606i | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 27.e | even | 9 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 729.2.e.g | 6 | |
| 3.b | odd | 2 | 1 | 729.2.e.b | 6 | ||
| 9.c | even | 3 | 1 | 729.2.e.a | 6 | ||
| 9.c | even | 3 | 1 | 729.2.e.h | 6 | ||
| 9.d | odd | 6 | 1 | 729.2.e.c | 6 | ||
| 9.d | odd | 6 | 1 | 729.2.e.i | 6 | ||
| 27.e | even | 9 | 1 | 243.2.a.e | ✓ | 3 | |
| 27.e | even | 9 | 2 | 243.2.c.f | 6 | ||
| 27.e | even | 9 | 1 | 729.2.e.a | 6 | ||
| 27.e | even | 9 | 1 | inner | 729.2.e.g | 6 | |
| 27.e | even | 9 | 1 | 729.2.e.h | 6 | ||
| 27.f | odd | 18 | 1 | 243.2.a.f | yes | 3 | |
| 27.f | odd | 18 | 2 | 243.2.c.e | 6 | ||
| 27.f | odd | 18 | 1 | 729.2.e.b | 6 | ||
| 27.f | odd | 18 | 1 | 729.2.e.c | 6 | ||
| 27.f | odd | 18 | 1 | 729.2.e.i | 6 | ||
| 108.j | odd | 18 | 1 | 3888.2.a.bd | 3 | ||
| 108.l | even | 18 | 1 | 3888.2.a.bk | 3 | ||
| 135.n | odd | 18 | 1 | 6075.2.a.bq | 3 | ||
| 135.p | even | 18 | 1 | 6075.2.a.bv | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 243.2.a.e | ✓ | 3 | 27.e | even | 9 | 1 | |
| 243.2.a.f | yes | 3 | 27.f | odd | 18 | 1 | |
| 243.2.c.e | 6 | 27.f | odd | 18 | 2 | ||
| 243.2.c.f | 6 | 27.e | even | 9 | 2 | ||
| 729.2.e.a | 6 | 9.c | even | 3 | 1 | ||
| 729.2.e.a | 6 | 27.e | even | 9 | 1 | ||
| 729.2.e.b | 6 | 3.b | odd | 2 | 1 | ||
| 729.2.e.b | 6 | 27.f | odd | 18 | 1 | ||
| 729.2.e.c | 6 | 9.d | odd | 6 | 1 | ||
| 729.2.e.c | 6 | 27.f | odd | 18 | 1 | ||
| 729.2.e.g | 6 | 1.a | even | 1 | 1 | trivial | |
| 729.2.e.g | 6 | 27.e | even | 9 | 1 | inner | |
| 729.2.e.h | 6 | 9.c | even | 3 | 1 | ||
| 729.2.e.h | 6 | 27.e | even | 9 | 1 | ||
| 729.2.e.i | 6 | 9.d | odd | 6 | 1 | ||
| 729.2.e.i | 6 | 27.f | odd | 18 | 1 | ||
| 3888.2.a.bd | 3 | 108.j | odd | 18 | 1 | ||
| 3888.2.a.bk | 3 | 108.l | even | 18 | 1 | ||
| 6075.2.a.bq | 3 | 135.n | odd | 18 | 1 | ||
| 6075.2.a.bv | 3 | 135.p | even | 18 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(729, [\chi])\):
|
\( T_{2}^{6} - 3T_{2}^{5} + 9T_{2}^{4} - 24T_{2}^{3} + 36T_{2}^{2} - 27T_{2} + 9 \)
|
|
\( T_{5}^{6} - 6T_{5}^{5} + 9T_{5}^{4} - 3T_{5}^{3} + 36T_{5}^{2} + 27T_{5} + 9 \)
|
|
\( T_{7}^{6} - 9T_{7}^{5} + 36T_{7}^{4} - 91T_{7}^{3} + 189T_{7}^{2} - 306T_{7} + 289 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 3 T^{5} + \cdots + 9 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} - 6 T^{5} + \cdots + 9 \)
|
| $7$ |
\( T^{6} - 9 T^{5} + \cdots + 289 \)
|
| $11$ |
\( T^{6} - 3 T^{5} + \cdots + 9 \)
|
| $13$ |
\( T^{6} - 9 T^{5} + \cdots + 289 \)
|
| $17$ |
\( (T^{2} - 3 T + 9)^{3} \)
|
| $19$ |
\( T^{6} - 3 T^{5} + \cdots + 1 \)
|
| $23$ |
\( T^{6} - 15 T^{5} + \cdots + 2601 \)
|
| $29$ |
\( T^{6} + 15 T^{5} + \cdots + 3249 \)
|
| $31$ |
\( T^{6} - 9 T^{5} + \cdots + 361 \)
|
| $37$ |
\( T^{6} - 3 T^{5} + \cdots + 1 \)
|
| $41$ |
\( T^{6} + 3 T^{5} + \cdots + 47961 \)
|
| $43$ |
\( T^{6} - 9 T^{5} + \cdots + 361 \)
|
| $47$ |
\( T^{6} + 15 T^{5} + \cdots + 71289 \)
|
| $53$ |
\( (T^{3} + 18 T^{2} + \cdots + 81)^{2} \)
|
| $59$ |
\( T^{6} - 6 T^{5} + \cdots + 103041 \)
|
| $61$ |
\( T^{6} - 18 T^{5} + \cdots + 2809 \)
|
| $67$ |
\( T^{6} + 9 T^{5} + \cdots + 11881 \)
|
| $71$ |
\( T^{6} + 9 T^{5} + \cdots + 998001 \)
|
| $73$ |
\( T^{6} + 6 T^{5} + \cdots + 157609 \)
|
| $79$ |
\( T^{6} - 9 T^{5} + \cdots + 2809 \)
|
| $83$ |
\( T^{6} - 21 T^{5} + \cdots + 2601 \)
|
| $89$ |
\( T^{6} + 189 T^{4} + \cdots + 998001 \)
|
| $97$ |
\( T^{6} - 36 T^{5} + \cdots + 361 \)
|
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