Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [81,10,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, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("81.28");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 81 = 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| 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: | \(41.7179027293\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{14})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 14x^{2} + 196 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{2} \) |
| 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
\( x^{4} + 14x^{2} + 196 \)
:
| \(\beta_{1}\) | \(=\) |
\( 6\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 14 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{3} ) / 7 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( 14\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 7\beta_{3} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(\beta_{2}\) |
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 |
|
−11.2250 | − | 19.4422i | 0 | 4.00000 | − | 6.92820i | 246.949 | − | 427.729i | 0 | 381.500 | + | 660.777i | −11674.0 | 0 | −11088.0 | ||||||||||||||||||||||
| 28.2 | 11.2250 | + | 19.4422i | 0 | 4.00000 | − | 6.92820i | −246.949 | + | 427.729i | 0 | 381.500 | + | 660.777i | 11674.0 | 0 | −11088.0 | |||||||||||||||||||||||
| 55.1 | −11.2250 | + | 19.4422i | 0 | 4.00000 | + | 6.92820i | 246.949 | + | 427.729i | 0 | 381.500 | − | 660.777i | −11674.0 | 0 | −11088.0 | |||||||||||||||||||||||
| 55.2 | 11.2250 | − | 19.4422i | 0 | 4.00000 | + | 6.92820i | −246.949 | − | 427.729i | 0 | 381.500 | − | 660.777i | 11674.0 | 0 | −11088.0 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 9.c | even | 3 | 1 | inner |
| 9.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 81.10.c.f | 4 | |
| 3.b | odd | 2 | 1 | inner | 81.10.c.f | 4 | |
| 9.c | even | 3 | 1 | 27.10.a.a | ✓ | 2 | |
| 9.c | even | 3 | 1 | inner | 81.10.c.f | 4 | |
| 9.d | odd | 6 | 1 | 27.10.a.a | ✓ | 2 | |
| 9.d | odd | 6 | 1 | inner | 81.10.c.f | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 27.10.a.a | ✓ | 2 | 9.c | even | 3 | 1 | |
| 27.10.a.a | ✓ | 2 | 9.d | odd | 6 | 1 | |
| 81.10.c.f | 4 | 1.a | even | 1 | 1 | trivial | |
| 81.10.c.f | 4 | 3.b | odd | 2 | 1 | inner | |
| 81.10.c.f | 4 | 9.c | even | 3 | 1 | inner | |
| 81.10.c.f | 4 | 9.d | odd | 6 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 504T_{2}^{2} + 254016 \)
acting on \(S_{10}^{\mathrm{new}}(81, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 504 T^{2} + 254016 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 59504772096 \)
|
| $7$ |
\( (T^{2} - 763 T + 582169)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 10\!\cdots\!76 \)
|
| $13$ |
\( (T^{2} - 73015 T + 5331190225)^{2} \)
|
| $17$ |
\( (T^{2} - 28304658144)^{2} \)
|
| $19$ |
\( (T + 598129)^{4} \)
|
| $23$ |
\( T^{4} + \cdots + 33\!\cdots\!96 \)
|
| $29$ |
\( T^{4} + \cdots + 46\!\cdots\!96 \)
|
| $31$ |
\( (T^{2} + \cdots + 3327340810000)^{2} \)
|
| $37$ |
\( (T - 14272175)^{4} \)
|
| $41$ |
\( T^{4} + \cdots + 79\!\cdots\!00 \)
|
| $43$ |
\( (T^{2} + \cdots + 60169559665216)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 92\!\cdots\!16 \)
|
| $53$ |
\( (T^{2} - 98034427661184)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 24\!\cdots\!56 \)
|
| $61$ |
\( (T^{2} + \cdots + 23\!\cdots\!49)^{2} \)
|
| $67$ |
\( (T^{2} + \cdots + 10\!\cdots\!61)^{2} \)
|
| $71$ |
\( (T^{2} - 11\!\cdots\!64)^{2} \)
|
| $73$ |
\( (T - 66463985)^{4} \)
|
| $79$ |
\( (T^{2} + \cdots + 13\!\cdots\!09)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 14\!\cdots\!76 \)
|
| $89$ |
\( (T^{2} - 11\!\cdots\!04)^{2} \)
|
| $97$ |
\( (T^{2} + \cdots + 55\!\cdots\!49)^{2} \)
|
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