Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [546,4,Mod(1,546)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(546, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("546.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 546.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: | \(32.2150428631\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 218x + 456 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 218x + 456 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 2\nu - 146 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 2\beta _1 + 146 \)
|
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 |
|
2.00000 | −3.00000 | 4.00000 | −14.1381 | −6.00000 | 7.00000 | 8.00000 | 9.00000 | −28.2761 | |||||||||||||||||||||||||||
| 1.2 | 2.00000 | −3.00000 | 4.00000 | −2.11461 | −6.00000 | 7.00000 | 8.00000 | 9.00000 | −4.22921 | ||||||||||||||||||||||||||||
| 1.3 | 2.00000 | −3.00000 | 4.00000 | 15.2527 | −6.00000 | 7.00000 | 8.00000 | 9.00000 | 30.5053 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(7\) | \(-1\) |
| \(13\) | \(1\) |
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 | 546.4.a.o | ✓ | 3 |
| 3.b | odd | 2 | 1 | 1638.4.a.x | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 546.4.a.o | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 1638.4.a.x | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} + T_{5}^{2} - 218T_{5} - 456 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(546))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{3} \)
|
| $3$ |
\( (T + 3)^{3} \)
|
| $5$ |
\( T^{3} + T^{2} + \cdots - 456 \)
|
| $7$ |
\( (T - 7)^{3} \)
|
| $11$ |
\( T^{3} - 63 T^{2} + \cdots + 21656 \)
|
| $13$ |
\( (T + 13)^{3} \)
|
| $17$ |
\( T^{3} + T^{2} + \cdots - 633276 \)
|
| $19$ |
\( T^{3} + 39 T^{2} + \cdots + 164992 \)
|
| $23$ |
\( T^{3} - 143 T^{2} + \cdots + 629328 \)
|
| $29$ |
\( T^{3} - 307 T^{2} + \cdots + 879972 \)
|
| $31$ |
\( T^{3} - 12 T^{2} + \cdots + 3331328 \)
|
| $37$ |
\( T^{3} - 429 T^{2} + \cdots + 4082564 \)
|
| $41$ |
\( T^{3} - 210 T^{2} + \cdots - 823968 \)
|
| $43$ |
\( T^{3} - 689 T^{2} + \cdots + 1604784 \)
|
| $47$ |
\( T^{3} - 582 T^{2} + \cdots - 2823296 \)
|
| $53$ |
\( T^{3} - 474 T^{2} + \cdots + 14769832 \)
|
| $59$ |
\( T^{3} + 74 T^{2} + \cdots - 2895744 \)
|
| $61$ |
\( T^{3} - 1045 T^{2} + \cdots - 36629172 \)
|
| $67$ |
\( T^{3} - 826 T^{2} + \cdots + 143634496 \)
|
| $71$ |
\( T^{3} - 630 T^{2} + \cdots - 4383360 \)
|
| $73$ |
\( T^{3} - 407 T^{2} + \cdots - 35460796 \)
|
| $79$ |
\( T^{3} - 942 T^{2} + \cdots - 4201600 \)
|
| $83$ |
\( T^{3} + 148 T^{2} + \cdots - 56728672 \)
|
| $89$ |
\( T^{3} + 940 T^{2} + \cdots - 594053376 \)
|
| $97$ |
\( T^{3} - 46 T^{2} + \cdots - 105032456 \)
|
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