Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [57,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("57.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 57 = 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(3.36310887033\) |
| 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: |
\( x^{4} - x^{3} - 27x^{2} - 13x + 88 \)
|
| 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
\( x^{4} - x^{3} - 27x^{2} - 13x + 88 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 13 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 4\nu^{2} - 15\nu + 28 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 13 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + 4\beta_{2} + 23\beta _1 + 24 \)
|
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 |
|
−4.67783 | 3.00000 | 13.8821 | −10.4211 | −14.0335 | 5.73511 | −27.5153 | 9.00000 | 48.7482 | ||||||||||||||||||||||||||||||
| 1.2 | −0.605921 | 3.00000 | −7.63286 | 11.5013 | −1.81776 | 26.0275 | 9.47228 | 9.00000 | −6.96889 | |||||||||||||||||||||||||||||||
| 1.3 | 3.67253 | 3.00000 | 5.48749 | 9.72745 | 11.0176 | −21.1323 | −9.22726 | 9.00000 | 35.7244 | |||||||||||||||||||||||||||||||
| 1.4 | 4.61122 | 3.00000 | 13.2633 | −16.8076 | 13.8336 | 27.3697 | 24.2702 | 9.00000 | −77.5037 | |||||||||||||||||||||||||||||||
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 | 57.4.a.d | ✓ | 4 |
| 3.b | odd | 2 | 1 | 171.4.a.h | 4 | ||
| 4.b | odd | 2 | 1 | 912.4.a.t | 4 | ||
| 5.b | even | 2 | 1 | 1425.4.a.k | 4 | ||
| 19.b | odd | 2 | 1 | 1083.4.a.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 57.4.a.d | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 171.4.a.h | 4 | 3.b | odd | 2 | 1 | ||
| 912.4.a.t | 4 | 4.b | odd | 2 | 1 | ||
| 1083.4.a.d | 4 | 19.b | odd | 2 | 1 | ||
| 1425.4.a.k | 4 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 3T_{2}^{3} - 24T_{2}^{2} + 66T_{2} + 48 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(57))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 3 T^{3} + \cdots + 48 \)
|
| $3$ |
\( (T - 3)^{4} \)
|
| $5$ |
\( T^{4} + 6 T^{3} + \cdots + 19596 \)
|
| $7$ |
\( T^{4} - 38 T^{3} + \cdots - 86336 \)
|
| $11$ |
\( T^{4} - 3555 T^{2} + \cdots + 611496 \)
|
| $13$ |
\( T^{4} - 68 T^{3} + \cdots + 1448560 \)
|
| $17$ |
\( T^{4} - 30 T^{3} + \cdots + 12247500 \)
|
| $19$ |
\( (T - 19)^{4} \)
|
| $23$ |
\( T^{4} + 246 T^{3} + \cdots - 23258112 \)
|
| $29$ |
\( T^{4} - 6 T^{3} + \cdots + 266996640 \)
|
| $31$ |
\( T^{4} - 188 T^{3} + \cdots + 26116096 \)
|
| $37$ |
\( T^{4} + \cdots - 2908078592 \)
|
| $41$ |
\( T^{4} + \cdots + 5830938624 \)
|
| $43$ |
\( T^{4} + \cdots + 4295336224 \)
|
| $47$ |
\( T^{4} + \cdots - 2545654992 \)
|
| $53$ |
\( T^{4} + \cdots + 8557525344 \)
|
| $59$ |
\( T^{4} + \cdots - 15363391488 \)
|
| $61$ |
\( T^{4} + \cdots + 1866482620 \)
|
| $67$ |
\( T^{4} + \cdots + 108714503680 \)
|
| $71$ |
\( T^{4} + \cdots + 94181583360 \)
|
| $73$ |
\( T^{4} + \cdots - 61797409892 \)
|
| $79$ |
\( T^{4} + \cdots + 56420337664 \)
|
| $83$ |
\( T^{4} + \cdots + 526934353536 \)
|
| $89$ |
\( T^{4} + \cdots - 181099849920 \)
|
| $97$ |
\( T^{4} + \cdots - 73056760016 \)
|
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