Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4,30,Mod(1,4)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 30, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4.1");
S:= CuspForms(chi, 30);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4 = 2^{2} \) |
| Weight: | \( k \) | \(=\) | \( 30 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4.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: | \(21.3112169532\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 25292x - 1534799 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{24}\cdot 3^{5}\cdot 7^{2}\cdot 23 \) |
| 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
\( x^{3} - 25292x - 1534799 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -28416\nu^{2} + 2727168\nu + 479131648 ) / 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 23587584\nu^{2} - 1455313152\nu - 397718116352 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( 37\beta_{2} + 30713\beta_1 ) / 5982584832 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3551\beta_{2} + 1894939\beta _1 + 100874357047296 ) / 5982584832 \)
|
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 | −6.72479e6 | 0 | 2.46424e10 | 0 | −2.78978e12 | 0 | −2.34075e13 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −2.56147e6 | 0 | −2.17115e10 | 0 | 8.03274e11 | 0 | −6.20693e13 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 1.54253e7 | 0 | 1.10142e10 | 0 | 2.34705e12 | 0 | 1.69310e14 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
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 | 4.30.a.a | ✓ | 3 |
| 3.b | odd | 2 | 1 | 36.30.a.d | 3 | ||
| 4.b | odd | 2 | 1 | 16.30.a.e | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4.30.a.a | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 16.30.a.e | 3 | 4.b | odd | 2 | 1 | ||
| 36.30.a.d | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{30}^{\mathrm{new}}(\Gamma_0(4))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + \cdots - 26\!\cdots\!28 \)
|
| $5$ |
\( T^{3} + \cdots + 58\!\cdots\!00 \)
|
| $7$ |
\( T^{3} + \cdots + 52\!\cdots\!08 \)
|
| $11$ |
\( T^{3} + \cdots - 22\!\cdots\!00 \)
|
| $13$ |
\( T^{3} + \cdots - 81\!\cdots\!52 \)
|
| $17$ |
\( T^{3} + \cdots - 22\!\cdots\!76 \)
|
| $19$ |
\( T^{3} + \cdots - 48\!\cdots\!96 \)
|
| $23$ |
\( T^{3} + \cdots - 15\!\cdots\!64 \)
|
| $29$ |
\( T^{3} + \cdots + 49\!\cdots\!44 \)
|
| $31$ |
\( T^{3} + \cdots + 16\!\cdots\!04 \)
|
| $37$ |
\( T^{3} + \cdots - 16\!\cdots\!52 \)
|
| $41$ |
\( T^{3} + \cdots - 13\!\cdots\!96 \)
|
| $43$ |
\( T^{3} + \cdots + 21\!\cdots\!00 \)
|
| $47$ |
\( T^{3} + \cdots + 12\!\cdots\!44 \)
|
| $53$ |
\( T^{3} + \cdots - 10\!\cdots\!56 \)
|
| $59$ |
\( T^{3} + \cdots - 64\!\cdots\!68 \)
|
| $61$ |
\( T^{3} + \cdots - 60\!\cdots\!68 \)
|
| $67$ |
\( T^{3} + \cdots - 28\!\cdots\!12 \)
|
| $71$ |
\( T^{3} + \cdots - 15\!\cdots\!52 \)
|
| $73$ |
\( T^{3} + \cdots + 80\!\cdots\!68 \)
|
| $79$ |
\( T^{3} + \cdots - 68\!\cdots\!92 \)
|
| $83$ |
\( T^{3} + \cdots + 30\!\cdots\!76 \)
|
| $89$ |
\( T^{3} + \cdots + 37\!\cdots\!36 \)
|
| $97$ |
\( T^{3} + \cdots + 61\!\cdots\!28 \)
|
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