Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [232,4,Mod(1,232)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(232, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("232.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 232 = 2^{3} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 232.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: | \(13.6884431213\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.225792.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 18x^{2} + 18 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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
\( x^{4} - 18x^{2} + 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + \nu^{2} - 12\nu - 9 ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{3} + 36\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{2} - 18 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} + 2\beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{3} + 18 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -9\beta_{3} + 6\beta_{2} + 18\beta_1 ) / 2 \)
|
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 |
|
0 | −9.41882 | 0 | −7.91581 | 0 | 19.7535 | 0 | 61.7142 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.11264 | 0 | 6.64036 | 0 | −11.4151 | 0 | −25.7620 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 4.12732 | 0 | −2.08419 | 0 | −13.1705 | 0 | −9.96522 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 6.40414 | 0 | −16.6404 | 0 | −3.16792 | 0 | 14.0130 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(29\) | \(-1\) |
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 | 232.4.a.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | 2088.4.a.e | 4 | ||
| 4.b | odd | 2 | 1 | 464.4.a.k | 4 | ||
| 8.b | even | 2 | 1 | 1856.4.a.w | 4 | ||
| 8.d | odd | 2 | 1 | 1856.4.a.x | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 232.4.a.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 464.4.a.k | 4 | 4.b | odd | 2 | 1 | ||
| 1856.4.a.w | 4 | 8.b | even | 2 | 1 | ||
| 1856.4.a.x | 4 | 8.d | odd | 2 | 1 | ||
| 2088.4.a.e | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 74T_{3}^{2} + 168T_{3} + 277 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(232))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 74 T^{2} + \cdots + 277 \)
|
| $5$ |
\( T^{4} + 20 T^{3} + \cdots - 1823 \)
|
| $7$ |
\( T^{4} + 8 T^{3} + \cdots - 9408 \)
|
| $11$ |
\( T^{4} - 8 T^{3} + \cdots + 2296893 \)
|
| $13$ |
\( T^{4} + 60 T^{3} + \cdots - 4326887 \)
|
| $17$ |
\( T^{4} + 224 T^{3} + \cdots - 55312368 \)
|
| $19$ |
\( T^{4} + 200 T^{3} + \cdots - 18943728 \)
|
| $23$ |
\( T^{4} + 104 T^{3} + \cdots + 26110672 \)
|
| $29$ |
\( (T - 29)^{4} \)
|
| $31$ |
\( T^{4} - 424 T^{3} + \cdots - 147822387 \)
|
| $37$ |
\( T^{4} + \cdots - 8389147392 \)
|
| $41$ |
\( T^{4} + 264 T^{3} + \cdots + 796866624 \)
|
| $43$ |
\( T^{4} + \cdots - 6763463843 \)
|
| $47$ |
\( T^{4} + 480 T^{3} + \cdots - 46277307 \)
|
| $53$ |
\( T^{4} + \cdots + 48201739177 \)
|
| $59$ |
\( T^{4} + \cdots + 27370709968 \)
|
| $61$ |
\( T^{4} + \cdots + 1049490576 \)
|
| $67$ |
\( T^{4} + 80 T^{3} + \cdots - 715776 \)
|
| $71$ |
\( T^{4} + \cdots + 50595925968 \)
|
| $73$ |
\( T^{4} + \cdots + 4749558016 \)
|
| $79$ |
\( T^{4} + \cdots + 75686955061 \)
|
| $83$ |
\( T^{4} + \cdots + 17011084752 \)
|
| $89$ |
\( T^{4} + \cdots - 2795276736 \)
|
| $97$ |
\( T^{4} + \cdots + 10030146112 \)
|
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