Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [81,8,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, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("81.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 81 = 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| 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: | \(25.3031870642\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 342x^{2} - 352x + 2512 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{3} \) |
| Twist minimal: | yes |
| 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
\( x^{4} - x^{3} - 342x^{2} - 352x + 2512 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3\nu - 171 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 328\nu - 268 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3\beta _1 + 171 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + \beta_{2} + 331\beta _1 + 439 \)
|
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 |
|
−15.3080 | 0 | 106.336 | −53.3856 | 0 | 243.109 | 331.630 | 0 | 817.229 | ||||||||||||||||||||||||||||||
| 1.2 | 1.74846 | 0 | −124.943 | 361.634 | 0 | −1517.71 | −442.261 | 0 | 632.303 | |||||||||||||||||||||||||||||||
| 1.3 | 7.35947 | 0 | −73.8382 | −468.472 | 0 | 1311.69 | −1485.42 | 0 | −3447.70 | |||||||||||||||||||||||||||||||
| 1.4 | 21.2001 | 0 | 321.445 | 352.223 | 0 | 762.913 | 4101.05 | 0 | 7467.17 | |||||||||||||||||||||||||||||||
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 | 81.8.a.b | yes | 4 |
| 3.b | odd | 2 | 1 | 81.8.a.a | ✓ | 4 | |
| 9.c | even | 3 | 2 | 81.8.c.i | 8 | ||
| 9.d | odd | 6 | 2 | 81.8.c.j | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 81.8.a.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 81.8.a.b | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 81.8.c.i | 8 | 9.c | even | 3 | 2 | ||
| 81.8.c.j | 8 | 9.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 15T_{2}^{3} - 258T_{2}^{2} + 2880T_{2} - 4176 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(81))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 15 T^{3} + \cdots - 4176 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 3185622225 \)
|
| $7$ |
\( T^{4} + \cdots - 369228406016 \)
|
| $11$ |
\( T^{4} + \cdots + 362906493428736 \)
|
| $13$ |
\( T^{4} + \cdots - 709887768510599 \)
|
| $17$ |
\( T^{4} + \cdots + 35\!\cdots\!89 \)
|
| $19$ |
\( T^{4} + \cdots - 59\!\cdots\!96 \)
|
| $23$ |
\( T^{4} + \cdots - 52\!\cdots\!56 \)
|
| $29$ |
\( T^{4} + \cdots + 28\!\cdots\!89 \)
|
| $31$ |
\( T^{4} + \cdots - 75\!\cdots\!24 \)
|
| $37$ |
\( T^{4} + \cdots + 15\!\cdots\!49 \)
|
| $41$ |
\( T^{4} + \cdots - 69\!\cdots\!04 \)
|
| $43$ |
\( T^{4} + \cdots - 17\!\cdots\!16 \)
|
| $47$ |
\( T^{4} + \cdots - 25\!\cdots\!44 \)
|
| $53$ |
\( T^{4} + \cdots - 94\!\cdots\!04 \)
|
| $59$ |
\( T^{4} + \cdots - 10\!\cdots\!16 \)
|
| $61$ |
\( T^{4} + \cdots + 27\!\cdots\!21 \)
|
| $67$ |
\( T^{4} + \cdots - 25\!\cdots\!24 \)
|
| $71$ |
\( T^{4} + \cdots - 18\!\cdots\!24 \)
|
| $73$ |
\( T^{4} + \cdots + 12\!\cdots\!89 \)
|
| $79$ |
\( T^{4} + \cdots + 40\!\cdots\!44 \)
|
| $83$ |
\( T^{4} + \cdots - 10\!\cdots\!84 \)
|
| $89$ |
\( T^{4} + \cdots - 34\!\cdots\!79 \)
|
| $97$ |
\( T^{4} + \cdots - 34\!\cdots\!24 \)
|
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