Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1,48,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, 48, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1.1");
S:= CuspForms(chi, 48);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1 \) |
| Weight: | \( k \) | \(=\) | \( 48 \) |
| 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: | \(13.9907662655\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{4} - x^{3} - 832803191366x^{2} + 3710135215485780x + 13175318942671469337000 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{20}\cdot 3^{7}\cdot 5^{3}\cdot 7^{2} \) |
| 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
\( x^{4} - x^{3} - 832803191366x^{2} + 3710135215485780x + 13175318942671469337000 \)
:
| \(\beta_{1}\) | \(=\) |
\( 24\nu - 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 60053\nu^{2} - 800757750008\nu - 22223988026907900 ) / 514752 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -485\nu^{3} + 267371447\nu^{2} + 390350086376920\nu - 112683253510933193364 ) / 514752 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 6 ) / 24 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 485\beta_{2} - 160480\beta _1 + 239847319113552 ) / 576 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -60053\beta_{3} + 267371447\beta_{2} + 19227823305632\beta _1 - 1602418642111186704 ) / 576 \)
|
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 |
|
−2.02978e7 | −2.01642e10 | 2.71261e14 | −3.11682e16 | 4.09288e17 | −1.26592e20 | −2.64934e21 | −2.61822e22 | 6.32644e23 | ||||||||||||||||||||||||||||||
| 1.2 | −1.54692e6 | 1.52034e11 | −1.38345e14 | 4.23962e16 | −2.35185e17 | −3.90714e19 | 4.31719e20 | −3.47448e21 | −6.55837e22 | |||||||||||||||||||||||||||||||
| 1.3 | 4.55092e6 | −2.21918e11 | −1.20027e14 | −3.08341e16 | −1.00993e18 | 1.11073e20 | −1.18672e21 | 2.26588e22 | −1.40324e23 | |||||||||||||||||||||||||||||||
| 1.4 | 2.30793e7 | 1.28510e11 | 3.91917e14 | −1.15086e16 | 2.96592e18 | 1.54211e19 | 5.79706e21 | −1.00741e22 | −2.65611e23 | |||||||||||||||||||||||||||||||
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 | 1.48.a.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 9.48.a.c | 4 | ||
| 4.b | odd | 2 | 1 | 16.48.a.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.48.a.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 9.48.a.c | 4 | 3.b | odd | 2 | 1 | ||
| 16.48.a.d | 4 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{48}^{\mathrm{new}}(\Gamma_0(1))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + \cdots + 32\!\cdots\!56 \)
|
| $3$ |
\( T^{4} + \cdots + 87\!\cdots\!36 \)
|
| $5$ |
\( T^{4} + \cdots - 46\!\cdots\!00 \)
|
| $7$ |
\( T^{4} + \cdots + 84\!\cdots\!96 \)
|
| $11$ |
\( T^{4} + \cdots - 18\!\cdots\!44 \)
|
| $13$ |
\( T^{4} + \cdots + 10\!\cdots\!76 \)
|
| $17$ |
\( T^{4} + \cdots - 14\!\cdots\!24 \)
|
| $19$ |
\( T^{4} + \cdots + 37\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots + 56\!\cdots\!16 \)
|
| $29$ |
\( T^{4} + \cdots - 69\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots + 13\!\cdots\!36 \)
|
| $37$ |
\( T^{4} + \cdots - 15\!\cdots\!64 \)
|
| $41$ |
\( T^{4} + \cdots + 89\!\cdots\!76 \)
|
| $43$ |
\( T^{4} + \cdots - 57\!\cdots\!04 \)
|
| $47$ |
\( T^{4} + \cdots + 84\!\cdots\!16 \)
|
| $53$ |
\( T^{4} + \cdots + 10\!\cdots\!36 \)
|
| $59$ |
\( T^{4} + \cdots - 14\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots - 47\!\cdots\!44 \)
|
| $67$ |
\( T^{4} + \cdots - 23\!\cdots\!24 \)
|
| $71$ |
\( T^{4} + \cdots - 15\!\cdots\!04 \)
|
| $73$ |
\( T^{4} + \cdots - 60\!\cdots\!84 \)
|
| $79$ |
\( T^{4} + \cdots + 29\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots - 10\!\cdots\!44 \)
|
| $89$ |
\( T^{4} + \cdots - 18\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots - 75\!\cdots\!84 \)
|
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