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: | 3.3.1772.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 12x + 8 \)
|
| 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
\( x^{3} - x^{2} - 12x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} + \nu - 8 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{2} + 5\nu + 6 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{2} + 5\beta _1 + 23 ) / 3 \)
|
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 |
|
−2.58488 | 0 | −1.31841 | 5.00000 | 0 | −22.8935 | 24.0869 | 0 | −12.9244 | |||||||||||||||||||||||||||
| 1.2 | 2.12612 | 0 | −3.47962 | 5.00000 | 0 | 30.7000 | −24.4070 | 0 | 10.6306 | ||||||||||||||||||||||||||||
| 1.3 | 5.45876 | 0 | 21.7980 | 5.00000 | 0 | −11.8065 | 75.3201 | 0 | 27.2938 | ||||||||||||||||||||||||||||
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.h | yes | 3 |
| 3.b | odd | 2 | 1 | 135.4.a.e | ✓ | 3 | |
| 4.b | odd | 2 | 1 | 2160.4.a.bq | 3 | ||
| 5.b | even | 2 | 1 | 675.4.a.p | 3 | ||
| 5.c | odd | 4 | 2 | 675.4.b.n | 6 | ||
| 9.c | even | 3 | 2 | 405.4.e.q | 6 | ||
| 9.d | odd | 6 | 2 | 405.4.e.v | 6 | ||
| 12.b | even | 2 | 1 | 2160.4.a.bi | 3 | ||
| 15.d | odd | 2 | 1 | 675.4.a.s | 3 | ||
| 15.e | even | 4 | 2 | 675.4.b.m | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 135.4.a.e | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 135.4.a.h | yes | 3 | 1.a | even | 1 | 1 | trivial |
| 405.4.e.q | 6 | 9.c | even | 3 | 2 | ||
| 405.4.e.v | 6 | 9.d | odd | 6 | 2 | ||
| 675.4.a.p | 3 | 5.b | even | 2 | 1 | ||
| 675.4.a.s | 3 | 15.d | odd | 2 | 1 | ||
| 675.4.b.m | 6 | 15.e | even | 4 | 2 | ||
| 675.4.b.n | 6 | 5.c | odd | 4 | 2 | ||
| 2160.4.a.bi | 3 | 12.b | even | 2 | 1 | ||
| 2160.4.a.bq | 3 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 5T_{2}^{2} - 8T_{2} + 30 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(135))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 5 T^{2} + \cdots + 30 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( (T - 5)^{3} \)
|
| $7$ |
\( T^{3} + 4 T^{2} + \cdots - 8298 \)
|
| $11$ |
\( T^{3} - 5 T^{2} + \cdots + 31260 \)
|
| $13$ |
\( T^{3} - 7 T^{2} + \cdots - 6425 \)
|
| $17$ |
\( T^{3} - 155 T^{2} + \cdots - 41760 \)
|
| $19$ |
\( T^{3} + 50 T^{2} + \cdots - 368012 \)
|
| $23$ |
\( T^{3} - 285 T^{2} + \cdots + 553500 \)
|
| $29$ |
\( T^{3} - 115 T^{2} + \cdots + 6440340 \)
|
| $31$ |
\( T^{3} + 115 T^{2} + \cdots - 938304 \)
|
| $37$ |
\( T^{3} + 384 T^{2} + \cdots - 22667198 \)
|
| $41$ |
\( T^{3} - 580 T^{2} + \cdots - 3917280 \)
|
| $43$ |
\( T^{3} + 797 T^{2} + \cdots - 5357936 \)
|
| $47$ |
\( T^{3} + 145 T^{2} + \cdots + 14388240 \)
|
| $53$ |
\( T^{3} + 400 T^{2} + \cdots - 12658320 \)
|
| $59$ |
\( T^{3} - 380 T^{2} + \cdots + 5205120 \)
|
| $61$ |
\( T^{3} + 152 T^{2} + \cdots - 5069066 \)
|
| $67$ |
\( T^{3} - 2 T^{2} + \cdots + 20769300 \)
|
| $71$ |
\( T^{3} - 40 T^{2} + \cdots + 216071280 \)
|
| $73$ |
\( T^{3} + 980 T^{2} + \cdots + 16447954 \)
|
| $79$ |
\( T^{3} - 1013 T^{2} + \cdots + 90596925 \)
|
| $83$ |
\( T^{3} - 270 T^{2} + \cdots + 84539160 \)
|
| $89$ |
\( T^{3} - 1020 T^{2} + \cdots + 125064000 \)
|
| $97$ |
\( T^{3} - 720 T^{2} + \cdots + 27430558 \)
|
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