Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [135,4,Mod(1,135)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(135, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("135.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 135 = 3^{3} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 135.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: | \(7.96525785077\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.5637.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 23x + 6 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 3 \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 23x + 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 15 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 15 \)
|
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 |
|
−5.20067 | 0 | 19.0470 | 5.00000 | 0 | 24.4013 | −57.4517 | 0 | −26.0034 | |||||||||||||||||||||||||||
| 1.2 | −0.258712 | 0 | −7.93307 | 5.00000 | 0 | 14.5174 | 4.12208 | 0 | −1.29356 | ||||||||||||||||||||||||||||
| 1.3 | 4.45938 | 0 | 11.8861 | 5.00000 | 0 | 5.08123 | 17.3296 | 0 | 22.2969 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( -1 \) |
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 | 135.4.a.f | ✓ | 3 |
| 3.b | odd | 2 | 1 | 135.4.a.g | yes | 3 | |
| 4.b | odd | 2 | 1 | 2160.4.a.bm | 3 | ||
| 5.b | even | 2 | 1 | 675.4.a.r | 3 | ||
| 5.c | odd | 4 | 2 | 675.4.b.l | 6 | ||
| 9.c | even | 3 | 2 | 405.4.e.t | 6 | ||
| 9.d | odd | 6 | 2 | 405.4.e.r | 6 | ||
| 12.b | even | 2 | 1 | 2160.4.a.be | 3 | ||
| 15.d | odd | 2 | 1 | 675.4.a.q | 3 | ||
| 15.e | even | 4 | 2 | 675.4.b.k | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 135.4.a.f | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 135.4.a.g | yes | 3 | 3.b | odd | 2 | 1 | |
| 405.4.e.r | 6 | 9.d | odd | 6 | 2 | ||
| 405.4.e.t | 6 | 9.c | even | 3 | 2 | ||
| 675.4.a.q | 3 | 15.d | odd | 2 | 1 | ||
| 675.4.a.r | 3 | 5.b | even | 2 | 1 | ||
| 675.4.b.k | 6 | 15.e | even | 4 | 2 | ||
| 675.4.b.l | 6 | 5.c | odd | 4 | 2 | ||
| 2160.4.a.be | 3 | 12.b | even | 2 | 1 | ||
| 2160.4.a.bm | 3 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} + T_{2}^{2} - 23T_{2} - 6 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(135))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + T^{2} - 23T - 6 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( (T - 5)^{3} \)
|
| $7$ |
\( T^{3} - 44 T^{2} + \cdots - 1800 \)
|
| $11$ |
\( T^{3} - 38 T^{2} + \cdots + 83280 \)
|
| $13$ |
\( T^{3} - 28 T^{2} + \cdots + 100120 \)
|
| $17$ |
\( T^{3} + 19 T^{2} + \cdots - 553887 \)
|
| $19$ |
\( T^{3} - 187 T^{2} + \cdots + 525871 \)
|
| $23$ |
\( T^{3} + 81 T^{2} + \cdots - 2043981 \)
|
| $29$ |
\( T^{3} - 160 T^{2} + \cdots + 7892760 \)
|
| $31$ |
\( T^{3} - 227 T^{2} + \cdots - 246321 \)
|
| $37$ |
\( T^{3} - 78 T^{2} + \cdots + 13637080 \)
|
| $41$ |
\( T^{3} + 338 T^{2} + \cdots - 12116640 \)
|
| $43$ |
\( T^{3} - 22 T^{2} + \cdots + 18464560 \)
|
| $47$ |
\( T^{3} + 472 T^{2} + \cdots - 283200 \)
|
| $53$ |
\( T^{3} - 521 T^{2} + \cdots + 939789 \)
|
| $59$ |
\( T^{3} - 140 T^{2} + \cdots + 34131480 \)
|
| $61$ |
\( T^{3} - 595 T^{2} + \cdots + 1782607 \)
|
| $67$ |
\( T^{3} - 878 T^{2} + \cdots + 11295000 \)
|
| $71$ |
\( T^{3} + 602 T^{2} + \cdots - 280550880 \)
|
| $73$ |
\( T^{3} - 1294 T^{2} + \cdots + 404091280 \)
|
| $79$ |
\( T^{3} - 629 T^{2} + \cdots - 2010303 \)
|
| $83$ |
\( T^{3} + 1287 T^{2} + \cdots - 346404411 \)
|
| $89$ |
\( T^{3} - 2154 T^{2} + \cdots - 74325600 \)
|
| $97$ |
\( T^{3} - 1392 T^{2} + \cdots - 63595520 \)
|
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