Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1,58,Mod(1,1)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1, base_ring=CyclotomicField(1))
chi = DirichletCharacter(H, H._module([]))
N = Newforms(chi, 58, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1.1");
S:= CuspForms(chi, 58);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1 \) |
| Weight: | \( k \) | \(=\) | \( 58 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1.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: | \(20.5766433651\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 20682206675887x^{2} + 1182366456513663853x + 45927816189452762789055234 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{25}\cdot 3^{10}\cdot 5^{2}\cdot 7\cdot 19 \) |
| 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
\( x^{4} - x^{3} - 20682206675887x^{2} + 1182366456513663853x + 45927816189452762789055234 \)
:
| \(\beta_{1}\) | \(=\) |
\( 144\nu - 36 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -27\nu^{3} + 4965084\nu^{2} + 438929828931105\nu - 75287058387744845106 ) / 12043136 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 8235\nu^{3} + 123348883428\nu^{2} - 123166652676786513\nu - 1268261056268698604351598 ) / 6021568 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 36 ) / 144 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 610\beta_{2} - 12347911\beta _1 + 214433118815601600 ) / 20736 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 45973\beta_{3} - 2284238582\beta_{2} + 584672101395737\beta _1 - 4596960370505569210512 ) / 5184 \)
|
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 |
|
−6.73315e8 | 5.51192e13 | 3.09237e17 | −1.13437e20 | −3.71126e22 | 6.16816e23 | −1.11179e26 | 1.46809e27 | 7.63789e28 | ||||||||||||||||||||||||||||||
| 1.2 | −2.78145e8 | −3.57694e13 | −6.67506e16 | 2.54023e19 | 9.94908e21 | 4.82921e23 | 5.86512e25 | −2.90592e26 | −7.06551e27 | |||||||||||||||||||||||||||||||
| 1.3 | 1.80177e8 | 4.51574e13 | −1.11652e17 | 3.84628e19 | 8.13632e21 | −1.87845e24 | −4.60832e25 | 4.69152e26 | 6.93010e27 | |||||||||||||||||||||||||||||||
| 1.4 | 5.53538e8 | −2.70314e13 | 1.62289e17 | −5.73891e19 | −1.49629e22 | 1.73157e24 | 1.00602e25 | −8.39347e26 | −3.17671e28 | |||||||||||||||||||||||||||||||
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 | 1.58.a.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 9.58.a.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.58.a.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 9.58.a.b | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{58}^{\mathrm{new}}(\Gamma_0(1))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + \cdots + 18\!\cdots\!16 \)
|
| $3$ |
\( T^{4} + \cdots + 24\!\cdots\!16 \)
|
| $5$ |
\( T^{4} + \cdots + 63\!\cdots\!00 \)
|
| $7$ |
\( T^{4} + \cdots - 96\!\cdots\!04 \)
|
| $11$ |
\( T^{4} + \cdots - 10\!\cdots\!44 \)
|
| $13$ |
\( T^{4} + \cdots - 52\!\cdots\!64 \)
|
| $17$ |
\( T^{4} + \cdots + 91\!\cdots\!56 \)
|
| $19$ |
\( T^{4} + \cdots - 44\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots + 15\!\cdots\!56 \)
|
| $29$ |
\( T^{4} + \cdots - 51\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots - 69\!\cdots\!64 \)
|
| $37$ |
\( T^{4} + \cdots - 47\!\cdots\!24 \)
|
| $41$ |
\( T^{4} + \cdots - 18\!\cdots\!24 \)
|
| $43$ |
\( T^{4} + \cdots - 59\!\cdots\!04 \)
|
| $47$ |
\( T^{4} + \cdots - 36\!\cdots\!64 \)
|
| $53$ |
\( T^{4} + \cdots - 30\!\cdots\!84 \)
|
| $59$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 11\!\cdots\!56 \)
|
| $67$ |
\( T^{4} + \cdots + 62\!\cdots\!56 \)
|
| $71$ |
\( T^{4} + \cdots - 62\!\cdots\!04 \)
|
| $73$ |
\( T^{4} + \cdots - 21\!\cdots\!44 \)
|
| $79$ |
\( T^{4} + \cdots + 35\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots - 26\!\cdots\!24 \)
|
| $89$ |
\( T^{4} + \cdots + 60\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 59\!\cdots\!36 \)
|
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