Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [728,2,Mod(1,728)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(728, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("728.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 728 = 2^{3} \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 728.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: | \(5.81310926715\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.64268.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 12x^{2} + 6x + 32 \)
|
| 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
\( x^{4} - x^{3} - 12x^{2} + 6x + 32 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 6\nu + 2 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - \nu - 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta _1 + 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 7\beta _1 + 4 \)
|
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 |
|
0 | −2.65932 | 0 | −2.96141 | 0 | −1.00000 | 0 | 4.07200 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.72110 | 0 | 3.13311 | 0 | −1.00000 | 0 | −0.0378271 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 2.19455 | 0 | −1.70715 | 0 | −1.00000 | 0 | 1.81604 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 3.18587 | 0 | 3.53544 | 0 | −1.00000 | 0 | 7.14978 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(1\) |
| \(13\) | \(1\) |
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 | 728.2.a.i | ✓ | 4 |
| 3.b | odd | 2 | 1 | 6552.2.a.br | 4 | ||
| 4.b | odd | 2 | 1 | 1456.2.a.v | 4 | ||
| 7.b | odd | 2 | 1 | 5096.2.a.s | 4 | ||
| 8.b | even | 2 | 1 | 5824.2.a.cb | 4 | ||
| 8.d | odd | 2 | 1 | 5824.2.a.ce | 4 | ||
| 13.b | even | 2 | 1 | 9464.2.a.z | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 728.2.a.i | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1456.2.a.v | 4 | 4.b | odd | 2 | 1 | ||
| 5096.2.a.s | 4 | 7.b | odd | 2 | 1 | ||
| 5824.2.a.cb | 4 | 8.b | even | 2 | 1 | ||
| 5824.2.a.ce | 4 | 8.d | odd | 2 | 1 | ||
| 6552.2.a.br | 4 | 3.b | odd | 2 | 1 | ||
| 9464.2.a.z | 4 | 13.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - T_{3}^{3} - 12T_{3}^{2} + 6T_{3} + 32 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(728))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - T^{3} + \cdots + 32 \)
|
| $5$ |
\( T^{4} - 2 T^{3} + \cdots + 56 \)
|
| $7$ |
\( (T + 1)^{4} \)
|
| $11$ |
\( T^{4} + T^{3} + \cdots + 488 \)
|
| $13$ |
\( (T + 1)^{4} \)
|
| $17$ |
\( T^{4} - 16 T^{3} + \cdots + 32 \)
|
| $19$ |
\( T^{4} + 10 T^{3} + \cdots - 784 \)
|
| $23$ |
\( T^{4} - 7 T^{3} + \cdots - 64 \)
|
| $29$ |
\( T^{4} - 4 T^{3} + \cdots - 568 \)
|
| $31$ |
\( T^{4} + T^{3} - 27 T^{2} + \cdots - 8 \)
|
| $37$ |
\( T^{4} - 5 T^{3} + \cdots - 508 \)
|
| $41$ |
\( T^{4} - 19 T^{3} + \cdots - 2104 \)
|
| $43$ |
\( T^{4} - 14 T^{3} + \cdots - 16 \)
|
| $47$ |
\( T^{4} + 13 T^{3} + \cdots - 4952 \)
|
| $53$ |
\( T^{4} + 8 T^{3} + \cdots - 544 \)
|
| $59$ |
\( T^{4} + 26 T^{3} + \cdots - 9344 \)
|
| $61$ |
\( T^{4} + T^{3} + \cdots + 104 \)
|
| $67$ |
\( T^{4} - 5 T^{3} + \cdots + 4208 \)
|
| $71$ |
\( T^{4} + 18 T^{3} + \cdots + 2048 \)
|
| $73$ |
\( T^{4} - 7 T^{3} + \cdots - 166 \)
|
| $79$ |
\( T^{4} - 9 T^{3} + \cdots - 112 \)
|
| $83$ |
\( T^{4} - 12 T^{3} + \cdots + 1088 \)
|
| $89$ |
\( T^{4} - 4 T^{3} + \cdots + 5524 \)
|
| $97$ |
\( T^{4} - 25 T^{3} + \cdots + 526 \)
|
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