Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [731,2,Mod(560,731)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(731, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("731.560");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 731 = 17 \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 731.d (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(5.83706438776\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 20x^{6} + 129x^{4} + 323x^{2} + 256 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 20x^{6} + 129x^{4} + 323x^{2} + 256 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} - 12\nu^{5} - 9\nu^{3} + 85\nu ) / 24 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 7\nu^{7} + 16\nu^{6} + 108\nu^{5} + 288\nu^{4} + 423\nu^{3} + 1488\nu^{2} + 437\nu + 2096 ) / 96 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 7\nu^{7} - 16\nu^{6} + 108\nu^{5} - 288\nu^{4} + 423\nu^{3} - 1488\nu^{2} + 437\nu - 2096 ) / 96 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 7\nu^{7} + 48\nu^{6} + 108\nu^{5} + 768\nu^{4} + 423\nu^{3} + 3120\nu^{2} + 437\nu + 3120 ) / 96 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -5\nu^{7} - 84\nu^{5} - 381\nu^{3} - 463\nu ) / 24 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + 2\beta_{5} + 2\beta_{4} + 2\beta_{3} - 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{6} - \beta_{5} + 2\beta_{4} - 14\beta_{2} + 37 \)
|
| \(\nu^{5}\) | \(=\) |
\( -15\beta_{7} - 28\beta_{5} - 28\beta_{4} - 23\beta_{3} + 47\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 18\beta_{6} + 15\beta_{5} - 33\beta_{4} + 159\beta_{2} - 332 \)
|
| \(\nu^{7}\) | \(=\) |
\( 171\beta_{7} + 318\beta_{5} + 318\beta_{4} + 234\beta_{3} - 425\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/731\mathbb{Z}\right)^\times\).
| \(n\) | \(173\) | \(562\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 560.1 |
|
0 | − | 3.20222i | −2.00000 | − | 0.837930i | 0 | − | 2.87630i | 0 | −7.25421 | 0 | |||||||||||||||||||||||||||||||||||||||
| 560.2 | 0 | − | 2.15629i | −2.00000 | − | 2.88045i | 0 | − | 2.59528i | 0 | −1.64959 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 560.3 | 0 | − | 1.89948i | −2.00000 | 0.651103i | 0 | 2.29537i | 0 | −0.608027 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 560.4 | 0 | − | 1.21991i | −2.00000 | 3.81798i | 0 | − | 4.55222i | 0 | 1.51183 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 560.5 | 0 | 1.21991i | −2.00000 | − | 3.81798i | 0 | 4.55222i | 0 | 1.51183 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 560.6 | 0 | 1.89948i | −2.00000 | − | 0.651103i | 0 | − | 2.29537i | 0 | −0.608027 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 560.7 | 0 | 2.15629i | −2.00000 | 2.88045i | 0 | 2.59528i | 0 | −1.64959 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 560.8 | 0 | 3.20222i | −2.00000 | 0.837930i | 0 | 2.87630i | 0 | −7.25421 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 731.2.d.b | ✓ | 8 |
| 17.b | even | 2 | 1 | inner | 731.2.d.b | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 731.2.d.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 731.2.d.b | ✓ | 8 | 17.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{2}^{\mathrm{new}}(731, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 20 T^{6} + \cdots + 256 \)
|
| $5$ |
\( T^{8} + 24 T^{6} + \cdots + 36 \)
|
| $7$ |
\( T^{8} + 41 T^{6} + \cdots + 6084 \)
|
| $11$ |
\( T^{8} + 59 T^{6} + \cdots + 64 \)
|
| $13$ |
\( (T^{4} + 5 T^{3} + \cdots - 121)^{2} \)
|
| $17$ |
\( T^{8} + 3 T^{7} + \cdots + 83521 \)
|
| $19$ |
\( (T - 2)^{8} \)
|
| $23$ |
\( T^{8} + 143 T^{6} + \cdots + 937024 \)
|
| $29$ |
\( T^{8} + 96 T^{6} + \cdots + 9216 \)
|
| $31$ |
\( T^{8} + 147 T^{6} + \cdots + 46656 \)
|
| $37$ |
\( T^{8} + 152 T^{6} + \cdots + 76176 \)
|
| $41$ |
\( T^{8} + 255 T^{6} + \cdots + 46656 \)
|
| $43$ |
\( (T - 1)^{8} \)
|
| $47$ |
\( (T^{4} + 4 T^{3} + \cdots + 216)^{2} \)
|
| $53$ |
\( (T^{4} - 23 T^{3} + \cdots - 1623)^{2} \)
|
| $59$ |
\( (T^{4} + 8 T^{3} + \cdots + 108)^{2} \)
|
| $61$ |
\( T^{8} + 210 T^{6} + \cdots + 685584 \)
|
| $67$ |
\( (T^{4} + T^{3} - 183 T^{2} + \cdots + 2239)^{2} \)
|
| $71$ |
\( T^{8} + 186 T^{6} + \cdots + 2232036 \)
|
| $73$ |
\( T^{8} + 302 T^{6} + \cdots + 788544 \)
|
| $79$ |
\( T^{8} + 416 T^{6} + \cdots + 2214144 \)
|
| $83$ |
\( (T^{4} - 7 T^{3} + \cdots + 363)^{2} \)
|
| $89$ |
\( (T^{4} + 14 T^{3} + \cdots + 5808)^{2} \)
|
| $97$ |
\( T^{8} + 143 T^{6} + \cdots + 46656 \)
|
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