Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1,54,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, 54, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1.1");
S:= CuspForms(chi, 54);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1 \) |
| Weight: | \( k \) | \(=\) | \( 54 \) |
| 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: | \(17.7903107608\) |
| 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: |
\( x^{4} - x^{3} - 2315873743412x^{2} - 421178019174503472x + 612167648493870378955584 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{27}\cdot 3^{8}\cdot 5^{3}\cdot 7\cdot 13 \) |
| 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
\( x^{4} - x^{3} - 2315873743412x^{2} - 421178019174503472x + 612167648493870378955584 \)
:
| \(\beta_{1}\) | \(=\) |
\( 96\nu - 24 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 611437\nu^{2} + 1620402960440\nu - 392120600840009328 ) / 119308 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 187\nu^{3} + 435432545\nu^{2} - 452992885700072\nu - 563273827738686130416 ) / 59654 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 24 ) / 96 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 374\beta_{2} + 26188788\beta _1 + 10671546209644800 ) / 9216 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 611437\beta_{3} - 870865090\beta_{2} + 171571478170596\beta _1 + 2911198475853482464512 ) / 9216 \)
|
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 |
|
−1.38568e8 | −5.95414e12 | 1.01938e16 | −5.35900e18 | 8.25051e20 | 2.55363e22 | −1.64427e23 | 1.60685e25 | 7.42585e26 | ||||||||||||||||||||||||||||||
| 1.2 | −8.51381e7 | 6.54009e12 | −1.75871e15 | 2.26338e17 | −5.56810e20 | −3.24923e22 | 9.16589e23 | 2.33895e25 | −1.92700e25 | |||||||||||||||||||||||||||||||
| 1.3 | 2.58420e7 | −2.97907e12 | −8.33939e15 | 5.38136e18 | −7.69850e19 | 2.33436e22 | −4.48271e23 | −1.05084e25 | 1.39065e26 | |||||||||||||||||||||||||||||||
| 1.4 | 1.29387e8 | 1.34470e12 | 7.73392e15 | −4.81259e18 | 1.73988e20 | −1.66125e22 | −1.64746e23 | −1.75750e25 | −6.22689e26 | |||||||||||||||||||||||||||||||
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 | 1.54.a.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 9.54.a.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.54.a.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 9.54.a.b | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{54}^{\mathrm{new}}(\Gamma_0(1))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + \cdots + 39\!\cdots\!76 \)
|
| $3$ |
\( T^{4} + \cdots + 15\!\cdots\!76 \)
|
| $5$ |
\( T^{4} + \cdots + 31\!\cdots\!00 \)
|
| $7$ |
\( T^{4} + \cdots + 32\!\cdots\!96 \)
|
| $11$ |
\( T^{4} + \cdots + 36\!\cdots\!76 \)
|
| $13$ |
\( T^{4} + \cdots + 14\!\cdots\!56 \)
|
| $17$ |
\( T^{4} + \cdots - 32\!\cdots\!64 \)
|
| $19$ |
\( T^{4} + \cdots - 62\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots + 24\!\cdots\!36 \)
|
| $29$ |
\( T^{4} + \cdots + 28\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots + 31\!\cdots\!96 \)
|
| $37$ |
\( T^{4} + \cdots + 39\!\cdots\!16 \)
|
| $41$ |
\( T^{4} + \cdots - 72\!\cdots\!44 \)
|
| $43$ |
\( T^{4} + \cdots + 79\!\cdots\!96 \)
|
| $47$ |
\( T^{4} + \cdots + 82\!\cdots\!56 \)
|
| $53$ |
\( T^{4} + \cdots - 80\!\cdots\!24 \)
|
| $59$ |
\( T^{4} + \cdots - 37\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 18\!\cdots\!76 \)
|
| $67$ |
\( T^{4} + \cdots - 18\!\cdots\!64 \)
|
| $71$ |
\( T^{4} + \cdots + 83\!\cdots\!36 \)
|
| $73$ |
\( T^{4} + \cdots + 80\!\cdots\!36 \)
|
| $79$ |
\( T^{4} + \cdots + 39\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 36\!\cdots\!16 \)
|
| $89$ |
\( T^{4} + \cdots - 28\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 21\!\cdots\!56 \)
|
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