Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [81,6,Mod(1,81)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(81, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("81.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 81 = 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 81.a (trivial) |
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: | yes |
| Analytic conductor: | \(12.9910894049\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.4875021.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 32x^{2} - 3x + 226 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2\cdot 3^{3} \) |
| Twist minimal: | no (minimal twist has level 9) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 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
\( x^{4} - 32x^{2} - 3x + 226 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + \nu^{2} + 19\nu - 13 ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{3} - 11\nu^{2} - 11\nu + 170 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 19\nu^{2} - 19\nu + 301 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} + 3\beta_1 ) / 18 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} - \beta _1 + 96 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -11\beta_{3} + 19\beta_{2} + 27 ) / 9 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−7.54935 | 0 | 24.9927 | 87.4311 | 0 | −41.2065 | 52.9007 | 0 | −660.048 | ||||||||||||||||||||||||||||||
| 1.2 | −2.95347 | 0 | −23.2770 | −64.8509 | 0 | −160.190 | 163.259 | 0 | 191.535 | |||||||||||||||||||||||||||||||
| 1.3 | 3.57955 | 0 | −19.1868 | 8.10775 | 0 | 175.428 | −183.226 | 0 | 29.0221 | |||||||||||||||||||||||||||||||
| 1.4 | 9.92326 | 0 | 66.4711 | 47.3121 | 0 | −2.03091 | 342.066 | 0 | 469.490 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 81.6.a.d | 4 | |
| 3.b | odd | 2 | 1 | 81.6.a.c | 4 | ||
| 9.c | even | 3 | 2 | 27.6.c.a | 8 | ||
| 9.d | odd | 6 | 2 | 9.6.c.a | ✓ | 8 | |
| 36.f | odd | 6 | 2 | 432.6.i.c | 8 | ||
| 36.h | even | 6 | 2 | 144.6.i.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 9.6.c.a | ✓ | 8 | 9.d | odd | 6 | 2 | |
| 27.6.c.a | 8 | 9.c | even | 3 | 2 | ||
| 81.6.a.c | 4 | 3.b | odd | 2 | 1 | ||
| 81.6.a.d | 4 | 1.a | even | 1 | 1 | trivial | |
| 144.6.i.c | 8 | 36.h | even | 6 | 2 | ||
| 432.6.i.c | 8 | 36.f | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 3T_{2}^{3} - 84T_{2}^{2} + 72T_{2} + 792 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(81))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 3 T^{3} + \cdots + 792 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 78 T^{3} + \cdots - 2174976 \)
|
| $7$ |
\( T^{4} + 28 T^{3} + \cdots - 2351756 \)
|
| $11$ |
\( T^{4} - 444 T^{3} + \cdots - 120652839 \)
|
| $13$ |
\( T^{4} + \cdots + 3591331348 \)
|
| $17$ |
\( T^{4} + \cdots - 917747509932 \)
|
| $19$ |
\( T^{4} + \cdots + 1740240514672 \)
|
| $23$ |
\( T^{4} + \cdots + 2546088335604 \)
|
| $29$ |
\( T^{4} + \cdots + 29497949996724 \)
|
| $31$ |
\( T^{4} + \cdots + 687830198031760 \)
|
| $37$ |
\( T^{4} + \cdots - 24412884987584 \)
|
| $41$ |
\( T^{4} + \cdots + 341598000989025 \)
|
| $43$ |
\( T^{4} + \cdots + 848570491190257 \)
|
| $47$ |
\( T^{4} + \cdots - 31\!\cdots\!48 \)
|
| $53$ |
\( T^{4} + \cdots + 92\!\cdots\!92 \)
|
| $59$ |
\( T^{4} + \cdots - 91\!\cdots\!51 \)
|
| $61$ |
\( T^{4} + \cdots - 47\!\cdots\!48 \)
|
| $67$ |
\( T^{4} + \cdots + 73\!\cdots\!41 \)
|
| $71$ |
\( T^{4} + \cdots - 82\!\cdots\!84 \)
|
| $73$ |
\( T^{4} + \cdots - 27\!\cdots\!08 \)
|
| $79$ |
\( T^{4} + \cdots + 67\!\cdots\!16 \)
|
| $83$ |
\( T^{4} + \cdots - 37\!\cdots\!36 \)
|
| $89$ |
\( T^{4} + \cdots + 30\!\cdots\!60 \)
|
| $97$ |
\( T^{4} + \cdots + 51\!\cdots\!53 \)
|
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