Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [57,6,Mod(1,57)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(57, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("57.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 57 = 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 57.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: | \(9.14187772934\) |
| 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: |
\( x^{4} - x^{3} - 90x^{2} + 118x + 1412 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{4} - x^{3} - 90x^{2} + 118x + 1412 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + \nu^{2} - 58\nu - 10 ) / 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 7\nu^{2} - 52\nu - 286 ) / 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta_{2} - \beta _1 + 46 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{3} + 7\beta_{2} + 59\beta _1 - 36 \)
|
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 |
|
−8.73296 | −9.00000 | 44.2646 | −34.5891 | 78.5966 | 140.686 | −107.106 | 81.0000 | 302.065 | ||||||||||||||||||||||||||||||
| 1.2 | −3.66552 | −9.00000 | −18.5639 | 85.2217 | 32.9897 | −12.5103 | 185.343 | 81.0000 | −312.382 | |||||||||||||||||||||||||||||||
| 1.3 | 5.82184 | −9.00000 | 1.89382 | 23.6174 | −52.3966 | −250.612 | −175.273 | 81.0000 | 137.497 | |||||||||||||||||||||||||||||||
| 1.4 | 7.57664 | −9.00000 | 25.4055 | −82.2501 | −68.1898 | −19.5642 | −49.9639 | 81.0000 | −623.180 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(19\) | \( +1 \) |
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 | 57.6.a.f | ✓ | 4 |
| 3.b | odd | 2 | 1 | 171.6.a.j | 4 | ||
| 4.b | odd | 2 | 1 | 912.6.a.r | 4 | ||
| 19.b | odd | 2 | 1 | 1083.6.a.g | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 57.6.a.f | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 171.6.a.j | 4 | 3.b | odd | 2 | 1 | ||
| 912.6.a.r | 4 | 4.b | odd | 2 | 1 | ||
| 1083.6.a.g | 4 | 19.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - T_{2}^{3} - 90T_{2}^{2} + 118T_{2} + 1412 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(57))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - T^{3} + \cdots + 1412 \)
|
| $3$ |
\( (T + 9)^{4} \)
|
| $5$ |
\( T^{4} + 8 T^{3} + \cdots + 5726088 \)
|
| $7$ |
\( T^{4} + 142 T^{3} + \cdots - 8629440 \)
|
| $11$ |
\( T^{4} + \cdots + 8153412000 \)
|
| $13$ |
\( T^{4} + \cdots + 158810973888 \)
|
| $17$ |
\( T^{4} + \cdots + 322720049196 \)
|
| $19$ |
\( (T + 361)^{4} \)
|
| $23$ |
\( T^{4} + \cdots + 18375207565952 \)
|
| $29$ |
\( T^{4} + \cdots + 112750314245840 \)
|
| $31$ |
\( T^{4} + \cdots + 126908512695648 \)
|
| $37$ |
\( T^{4} + \cdots - 905306584227840 \)
|
| $41$ |
\( T^{4} + \cdots + 20\!\cdots\!88 \)
|
| $43$ |
\( T^{4} + \cdots + 364921961696464 \)
|
| $47$ |
\( T^{4} + \cdots - 81\!\cdots\!48 \)
|
| $53$ |
\( T^{4} + \cdots - 50\!\cdots\!96 \)
|
| $59$ |
\( T^{4} + \cdots - 83\!\cdots\!80 \)
|
| $61$ |
\( T^{4} + \cdots + 67\!\cdots\!28 \)
|
| $67$ |
\( T^{4} + \cdots - 12\!\cdots\!28 \)
|
| $71$ |
\( T^{4} + \cdots - 14\!\cdots\!00 \)
|
| $73$ |
\( T^{4} + \cdots - 25\!\cdots\!60 \)
|
| $79$ |
\( T^{4} + \cdots - 29\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots - 30\!\cdots\!52 \)
|
| $89$ |
\( T^{4} + \cdots - 66\!\cdots\!20 \)
|
| $97$ |
\( T^{4} + \cdots + 61\!\cdots\!40 \)
|
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