Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [133,2,Mod(1,133)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(133, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("133.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 133 = 7 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 133.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: | \(1.06201034688\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.229.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 4x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{3} - 4x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
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 |
|
−1.93543 | 1.25410 | 1.74590 | 1.68133 | −2.42723 | −1.00000 | 0.491797 | −1.42723 | −3.25410 | |||||||||||||||||||||||||||
| 1.2 | 1.46260 | 2.86081 | 0.139194 | −3.32340 | 4.18421 | −1.00000 | −2.72161 | 5.18421 | −4.86081 | ||||||||||||||||||||||||||||
| 1.3 | 2.47283 | −1.11491 | 4.11491 | −0.357926 | −2.75698 | −1.00000 | 5.22982 | −1.75698 | −0.885092 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(1\) |
| \(19\) | \(-1\) |
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 | 133.2.a.d | ✓ | 3 |
| 3.b | odd | 2 | 1 | 1197.2.a.k | 3 | ||
| 4.b | odd | 2 | 1 | 2128.2.a.p | 3 | ||
| 5.b | even | 2 | 1 | 3325.2.a.r | 3 | ||
| 7.b | odd | 2 | 1 | 931.2.a.k | 3 | ||
| 7.c | even | 3 | 2 | 931.2.f.l | 6 | ||
| 7.d | odd | 6 | 2 | 931.2.f.m | 6 | ||
| 8.b | even | 2 | 1 | 8512.2.a.bi | 3 | ||
| 8.d | odd | 2 | 1 | 8512.2.a.bp | 3 | ||
| 19.b | odd | 2 | 1 | 2527.2.a.f | 3 | ||
| 21.c | even | 2 | 1 | 8379.2.a.bo | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 133.2.a.d | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 931.2.a.k | 3 | 7.b | odd | 2 | 1 | ||
| 931.2.f.l | 6 | 7.c | even | 3 | 2 | ||
| 931.2.f.m | 6 | 7.d | odd | 6 | 2 | ||
| 1197.2.a.k | 3 | 3.b | odd | 2 | 1 | ||
| 2128.2.a.p | 3 | 4.b | odd | 2 | 1 | ||
| 2527.2.a.f | 3 | 19.b | odd | 2 | 1 | ||
| 3325.2.a.r | 3 | 5.b | even | 2 | 1 | ||
| 8379.2.a.bo | 3 | 21.c | even | 2 | 1 | ||
| 8512.2.a.bi | 3 | 8.b | even | 2 | 1 | ||
| 8512.2.a.bp | 3 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 2T_{2}^{2} - 4T_{2} + 7 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(133))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 2 T^{2} + \cdots + 7 \)
|
| $3$ |
\( T^{3} - 3T^{2} - T + 4 \)
|
| $5$ |
\( T^{3} + 2 T^{2} + \cdots - 2 \)
|
| $7$ |
\( (T + 1)^{3} \)
|
| $11$ |
\( T^{3} - 7 T^{2} + \cdots - 4 \)
|
| $13$ |
\( T^{3} + 2 T^{2} + \cdots - 2 \)
|
| $17$ |
\( T^{3} - 7 T^{2} + \cdots + 106 \)
|
| $19$ |
\( (T - 1)^{3} \)
|
| $23$ |
\( T^{3} - 14 T^{2} + \cdots - 56 \)
|
| $29$ |
\( T^{3} + 3 T^{2} + \cdots - 278 \)
|
| $31$ |
\( T^{3} + 11 T^{2} + \cdots + 16 \)
|
| $37$ |
\( T^{3} - 43T + 106 \)
|
| $41$ |
\( T^{3} + 7 T^{2} + \cdots - 998 \)
|
| $43$ |
\( T^{3} + 4 T^{2} + \cdots - 16 \)
|
| $47$ |
\( T^{3} - 8 T^{2} + \cdots - 16 \)
|
| $53$ |
\( T^{3} + T^{2} - 31T - 2 \)
|
| $59$ |
\( T^{3} + 10 T^{2} + \cdots - 124 \)
|
| $61$ |
\( T^{3} + 6 T^{2} + \cdots - 82 \)
|
| $67$ |
\( T^{3} + 3 T^{2} + \cdots - 188 \)
|
| $71$ |
\( T^{3} - 61T - 32 \)
|
| $73$ |
\( T^{3} - T^{2} + \cdots - 98 \)
|
| $79$ |
\( T^{3} + 4 T^{2} + \cdots + 32 \)
|
| $83$ |
\( T^{3} - 31 T^{2} + \cdots - 788 \)
|
| $89$ |
\( T^{3} + 28 T^{2} + \cdots - 1352 \)
|
| $97$ |
\( T^{3} + 30 T^{2} + \cdots + 482 \)
|
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