Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [81,8,Mod(28,81)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(81, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("81.28");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 81 = 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 81.c (of order \(3\), degree \(2\), 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: | \(25.3031870642\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{65})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 17x^{2} + 16x + 256 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | no (minimal twist has level 27) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} + 17x^{2} + 16x + 256 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 17\nu^{2} - 17\nu + 256 ) / 272 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 17\nu^{2} + 833\nu - 256 ) / 272 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{3} + 82 ) / 17 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 49\beta _1 - 50 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 17\beta_{3} - 82 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 28.1 |
|
−3.79669 | − | 6.57607i | 0 | 35.1702 | − | 60.9166i | 32.9066 | − | 56.9959i | 0 | 369.202 | + | 639.477i | −1506.08 | 0 | −499.745 | ||||||||||||||||||||||
| 28.2 | 8.29669 | + | 14.3703i | 0 | −73.6702 | + | 127.601i | 57.0934 | − | 98.8886i | 0 | −719.202 | − | 1245.70i | −320.924 | 0 | 1894.75 | |||||||||||||||||||||||
| 55.1 | −3.79669 | + | 6.57607i | 0 | 35.1702 | + | 60.9166i | 32.9066 | + | 56.9959i | 0 | 369.202 | − | 639.477i | −1506.08 | 0 | −499.745 | |||||||||||||||||||||||
| 55.2 | 8.29669 | − | 14.3703i | 0 | −73.6702 | − | 127.601i | 57.0934 | + | 98.8886i | 0 | −719.202 | + | 1245.70i | −320.924 | 0 | 1894.75 | |||||||||||||||||||||||
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 | 81.8.c.h | 4 | |
| 3.b | odd | 2 | 1 | 81.8.c.d | 4 | ||
| 9.c | even | 3 | 1 | 27.8.a.b | ✓ | 2 | |
| 9.c | even | 3 | 1 | inner | 81.8.c.h | 4 | |
| 9.d | odd | 6 | 1 | 27.8.a.e | yes | 2 | |
| 9.d | odd | 6 | 1 | 81.8.c.d | 4 | ||
| 36.f | odd | 6 | 1 | 432.8.a.j | 2 | ||
| 36.h | even | 6 | 1 | 432.8.a.q | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 27.8.a.b | ✓ | 2 | 9.c | even | 3 | 1 | |
| 27.8.a.e | yes | 2 | 9.d | odd | 6 | 1 | |
| 81.8.c.d | 4 | 3.b | odd | 2 | 1 | ||
| 81.8.c.d | 4 | 9.d | odd | 6 | 1 | ||
| 81.8.c.h | 4 | 1.a | even | 1 | 1 | trivial | |
| 81.8.c.h | 4 | 9.c | even | 3 | 1 | inner | |
| 432.8.a.j | 2 | 36.f | odd | 6 | 1 | ||
| 432.8.a.q | 2 | 36.h | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 9T_{2}^{3} + 207T_{2}^{2} + 1134T_{2} + 15876 \)
acting on \(S_{8}^{\mathrm{new}}(81, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 9 T^{3} + \cdots + 15876 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 180 T^{3} + \cdots + 56475225 \)
|
| $7$ |
\( T^{4} + \cdots + 1128109515625 \)
|
| $11$ |
\( T^{4} + \cdots + 865184866700625 \)
|
| $13$ |
\( T^{4} + \cdots + 46\!\cdots\!00 \)
|
| $17$ |
\( (T^{2} + 16416 T - 741669696)^{2} \)
|
| $19$ |
\( (T^{2} - 16024 T - 142216916)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 15\!\cdots\!36 \)
|
| $29$ |
\( T^{4} + \cdots + 26\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots + 81\!\cdots\!61 \)
|
| $37$ |
\( (T^{2} - 455620 T - 16337003900)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 69\!\cdots\!00 \)
|
| $43$ |
\( T^{4} + \cdots + 43\!\cdots\!00 \)
|
| $47$ |
\( T^{4} + \cdots + 16\!\cdots\!00 \)
|
| $53$ |
\( (T^{2} + \cdots + 1583691044571)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 41\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 30\!\cdots\!76 \)
|
| $67$ |
\( T^{4} + \cdots + 24\!\cdots\!00 \)
|
| $71$ |
\( (T^{2} + \cdots - 3541802241600)^{2} \)
|
| $73$ |
\( (T^{2} + \cdots - 3239699519975)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 36\!\cdots\!16 \)
|
| $83$ |
\( T^{4} + \cdots + 28\!\cdots\!81 \)
|
| $89$ |
\( (T^{2} + \cdots + 6381093451500)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 24\!\cdots\!25 \)
|
| show more | |
| show less |